I already suggested trying Linux on the existing computer (second post in this thread), and was basically shown the middle finger for doing so.

Are these games from your GOG library, or old retail versions?

Ah, Starchen has a good point as many old copy protections will hinder you from playing your games...

Your posts seem a bit confusing on what you want to achieve. You have a desktop PC and want to buy another one to play older games, is this right?

I can recomend a old lenovo/hp/dell with a Core2quad / AMD Phenom x4, or better yet, a 2ºgen i5, those CPU's run older and newer games without problem (preferably with ddr3 RAM) exemple HP elite 8300/ Dell optiplex 990/ Fujitsu e400.
I have a core2quad q6600 machine with win7 and a pentiumg3258 with win8.1, never had any GOG game that don't work. Some games require a bit of work to get the better out of them but out of the box all worked.

For the GPU it really depends on the computer you want, like a SFF and resolution you play newer games but a GTX 750ti / HD 7770 should play most older games without problem, there are Low profile versions and both support Win XP if you want. Nvidia dropped XP support on 10 series and AMD on R+ series.
Both support newer titles as well.

Lets get something out of the way, Hp Omen is just a brand (fancy leds yeahhh), the main pc components wich require drivers are usually not made by HP. Sometimes they rebrand a GPU and some motherboards require some drivers... With that said, there are no modern gaming GPU made to this day that don't have drivers to win7, most CPU´s can be made to work on win7 and most motherboard components that may require drivers, usually support win7 (i've yet to see a component that don't support).
If you really need/want win7 there are ways to do it, despite what HP said to you...

No one is getting your point because there are pretty much no Graphic cards that don´t have drivers for win7 or some Linux distro. Even newer cards, crap even future cards will probably support win7...

Since you don't want to disclose what components your computer have (being a HP Omen is not a spec...) search you GPU model here or
here

The thing is, Win10 is pretty "good" to auto search and install drivers but older OS you have to do it manually.
If you need any help to locate any driver, open device manager and check what drivers are due for installing, ask here and you will get some help...

As Trooper1270 posted, trying a Linux Os on a usb pen doesn't hurt your system, go to System 76 POP OS webpage and choose whatever GPU brand you have, flash it on a USB pen and try it...

Edit: messed up the links, click in the "here" word