With my current desktop PC having cooling issues and there being a security flaw in the integrated GPU whose mitigation apparently drastically hurts performance (using a Haswell CPU (i5-4670)), I think it might be time to think about buying or building a new one, and am wondering what advice you have.

What I will be using this for:
* Games like Bard's Tale 4, The Alliance Alive, and Hollow Knight. In general, it should be able to handle current generation console ports at settings that match the console version.
* Virtual machines, particularly for experimentation.
* Compiling software, including the Linux kernel and perhaps even something like Buildroot. (I don't think I'm going to go so far as to run Gentoo as the primary OS on the new machine.)
* Software development.
* Note that I will definitely not be running Windows on bare metal, so I don't need to budget for an OS (if custom built); the first thing I intend to do is install Linux (probably Debian testing).

Minimum requirements (I don't want the new PC to be a downgrade):
* CPU: Quad-core, minimum 3.4GHz (or equivalent to the i5-4670, but newer generation), must have hardware virtualization support (and I would *really* like Vt-d support here).
* GPU: Integrated works fine for me. One hard requirement is that the computer *must not* have a NVIDIA GPU, as I do not want to use a proprietary driver for the GPU.
* RAM: 16 GB. (Might 32GB be useful for my intended uses, particularly since currently much of that 16GB is taken up by open Chromium tabs?)
* Must have USB 3.0 (or better support).
* All hardware must be supported under Linux.
* Cooling must be functional enough to keep the CPU from throttling under heavy load.

Wishlist:
* Vt-d, as mentioned above.
* A USB-C port capable of providing enough power to run a Raspberry Pi 4 and function as a USB host. Bonus points if it could also act as a USB device, but I am not expecting that.

Budget: $1000 as cap, but paying less is preferable, of course.

So, any advice here?

Vt-d and Vt-x, or AMD equivalent (AMD-V and AMD-Vi), pretty much any non-entry level has those them.

Does power consumption and size matter to you?
Will you reuse your storage drives?
Need a lot of space?

Choosing a decent desktop with integrated grophics may be dificult at the moment, as most good value CPU's don't have them.
Not nVidia dedicated video cheap cards, are very lousy and very expensive to non existent.

With that in mind, my recomendation goes to a home built system based on AMD 2400G/3400G (4c8t with the best integrated graphics), for a low-cost oriented pc, for 500 to 600.

For a custom build more performance oriented with a budget around 700-1000, a Ryzen 2700(8c16t) and Rx570 seem to be unbeatable for your use case.

Other than that, a pre-built (choose your favorite brand) with a decent CPU, like the i7-8700 or 9700 non-k variants may be a good option.
Keep in mind Intel still holds a very high advantage with most game emulators.

As a quick note: if your i5- 4670 is still enough, maybe the option of add a dedicated GPU can somewhat revive your system with out a big expense. A Rx560 should be a good candidate, depending on your current setup. (the Rx550 is very expensive on my country for what it is)

Edit: while writing, completely forgot about the cooling issues (it probably can be repaired with some diy), but decided to leave the gpu upgrade bit.

Buy refurbished/off lease computers is my suggestion.

Good to know, since that was definitely not the case when I made my current pc. (In particular, the "k" model of the CPU II got didn't have Vt-d, but the non-k did.) * Power consumption and size matter to an extent, but I am not as picky as I would be with other devices (like a laptop or Raspberry Pi).
* I have not decided if I will reuse the storage drives. In any case, it appears that a 1 TB solid state drive can be had for like $110 (the costs have really come down over the years).
* I don't need a lot of space all the time; sometimes having extra space might be useful (if I have a reason to run a large VM, for example, or scratch space for buildroot), but I don't need it all the time. I would like something more recent than those, as new features get added to CPUs and other devices (and I might want to play around with some of them), and there's the whole issue that the computers mentioned at that link have the same generation CPU as mine, which has the drawback that the mitigation for the recent GPU vulnerability apparently comes with significant performance loss.

HOLY CRAP!!! Should be more prudent and carefully as I didn't know that.
I did check the CPU's on my recomendation list and all have those active.

The way I usually go about choosing parts is deciding the size/power, then choose the CPU, GPU and Ram, lastly adjusting all the required components.

Did a fast search on my go-to website. Keep in mind stated prices are in rounded Euros and might differ a little in your country.

CPU Ryzen 7 2700 8c16t - 150€
RAM 32Gb 8x4 GSkill F4-3000C16D-16GISB - 145€
Board MSI B450 Mortar Max m-ATX - 90€

Add a power supply like a Cooler Master MWE 450W - 45€
and a m-ATX case of your taste, like a simple Cooler Master Masterbox lite 3 for 50€

That yould leave you with ~500€ to sort storage and Video card, wich is required since the CPU has no integrated graphics.
A AMD Rx570 4Gb card ~135€ would be miles faster than any integrated graphics, probably in order of magnitude, in case you ever need for any upcoming game.

The choosen MB has indeed Usb-C connector but couldn't find the max current the port can output, it could be a Usb3.1 just with another connector (1 to 1.5 amps max). Raspi4 requires 3amps to be safe, probably ok with 2amps if not powering many pripherals, hats or display.
Contacting the board manufacturer would be a nice idea.

All parts seem to work flawlessly on Linux out of the box.

Just doing a bit of research, and put together a quick build on pcpartpicker.com just to get a rough idea of the cost. Using the Intel Core i3 CPU for its integrated graphics that are better than what I have (and not crippled by the mitigation I mentioned), and it apparently still performs better than my current i5, due to technology improving over the years.

I got a little greedy with the memory in this trial build, but for many other parts, just chose the cheapest one that was near the top of the list, as I am just trying to get an idea of how much things might cost, and I seem to have found something that costs around $620 in total.

Any thoughts? (Note that this is not even close to final; see this as a (very) rough draft.)

https://pcpartpicker.com/list/MjVTPn

(Also, anything else that I should add? Note that I am planning to install Linux and won't run Linux, so I don't need to pay for a Windows OS.)

I was just about to recommend a Ryzen build, since you'd end up cheaper, but then I noticed you mentioned Vt-d, and since I was clueless what it was, I searched around a bit. I'll be honest, I'm still pretty clueless about the subject, but maybe you'll find this useful: https://forum.level1techs.com/t/best-2019-motherboard-other-options-for-vt-d-linux/145117/15

As another poster mentioned, AMD calls their equivalent feature AMD-VI.

I am thinking of going Intel for better single-threaded performance (good for emulators and some games) and the integrated GPU being both well-supported in Linux and having such features as GVT-g (allows a VM to share the GPU with the host).

Incidentally, if I were building the computer as a purely work machine, AMD might be the better choice due to better multi-core performance (to my understanding), which would reduce compile time. (Any sufficiently large software project consists of many different components that can be built separately and then combined, so extra cores are a benefit there.)

The problem with the current generation of Intel processors, is that while they can run faster, they also run hotter. You'll need plenty of cooling power(expensive) in order to keep an Intel cold enough to maintain 5GHz.

If you're looking at emulators, I think intel still holds the edge, but if you consider the latest ryzen 3000 generation, that edge is under 5%. And most importantly the edge is only noticeable when you try to do extreme things, such as rendering the game internally at 4k, with multiple filters.

If you're interested in just playing games at 60fps 1080p, then Ryzen should be just as good as Intel.

However, every once in a while, I still see people complaining that their AMD video cards can cause problems in certain games. That is because while AMD's CPUs have spread enough that they can't be ignored anymore, the GPU market is still crowded by Nvidia. So there aren't enough developers optimizing emulators to take advantage of AMD GPUs.

When it comes to AMD-VI, I'm just as lost as I am about Vt-d. But I'm sure more knowledgeable people will leave their input.

Why would you limit yourself to a "worse" and more expensive CPU when Ryzen, since 1. gen, has proved yet again that they excel on performance per dollar, internal graphics, and has seriously good multi-core performance (which IS the future)?

Not only that - Intel has always limited their lower CPU by excluding support for ECC and virtualization. AMD has "always" (excluding a very few models) supported both in one way or the other.

Also to my knowledge, i3 is/was limited by having hyper-threading but not virtualization, and in i5 it was the other way around for some strange reason.

Every single AMD processor (since around the phenom days) has one or both AMD-Vi and AMD-Vi (I/O MMU virtualization) support, and supplied with a good mamaboard chipset and a fast memory (AMD scales much better with memory) you can't go wrong with this setup.

In synthetic tests we literally see only seconds between Intel and Amd, and why throw in many dollars on a few seconds (again in synthetic tests, not real world)... or a few more fps for that matter.

It seems to me that doing VM, coding and compiling are your main use so in my book single performance should only be a minor footnote, but of course, it's your choice.

Relevant:
http://techgenix.com/difference-between-amd-vintel-vt-x-and-amd-viintel-vt-d-188/

And that seems to be an article form 2011, so clearly the matter should be a non-issue for all modern CPUs.

This virtualization thru the graphics thing is actually new to me, and it seems AMD has something equivalent:
https://www.reddit.com/r/Amd/comments/dqwvql/intel_gvt_g_alternative_on_navi_cards/ (this time just a few months old ;) )

Too bad it's only for the more expensive server stuff... so yeah, I can understand it would be fun to experiment with that.

EDIT: The name is AMD MxGPU.

https://github.com/GPUOpen-LibrariesAndSDKs/MxGPU-Virtualization

Given the concerns about crippling security mitigations, I would recommend avoiding Intel.

Because of the 10nm fiasco, the fundamental CPU and graphics architecture of the latest Intel desktop chips hasn't changed since 2015. Yes, security fixes for known vulnerabilities have been included but there has been no redesign to fix the unsafe performance at any cost philosophy.

That's apparently not true these days. The Coffee Lake i3, for example, is a quad core without hyperthreading (much like the Haswell i5 I currently have), and according to https://ark.intel.com/content/www/us/en/ark/products/126688/intel-core-i3-8100-processor-6m-cache-3-60-ghz.html it supports vt-d. Basic vt-x support (hardware virtualization) is even found on low-end chips these days; my laptop's Celeron even supports it. (I actually even have a device with an ARM Cortex-A72 (a Raspberry Pi 4, specifically) that I've successfully gotten hardware virtualization to work on, but that device is not suited for playing x86 games.)

Then again, the whole vt-d on 4670 but not 4670K was strange back in the day.

Edit: Is there an AMD counterpart to ark.intel.com?