After ticking "Disable fullscreen optimizations" for HiddenEvil.exe in the Bin folder under Compatability the game boots every time (otherwise it was kinda random with one day it booting every time and the other not at all).
The intro movies work, the game also listens to the nGlide configurator (properly uses DirectX or Vulkan when I tell it to, 3Dfx logo can be disabled, resolution etc).

But after pressing NEW I get either a stuck screen (DirectX) or a black screen (Vulkan) and the game's Sprint Engine's CPU and disk usage drops to 0% and does nothing until I hit a button which causes the game to crash with the standard "has stopped working" error message where my only option is to Close program.

I've tried keeping everything on default in the configurator as well as using low resolutions. Run as administrator. XP SP2 compatability (changes DirectX behaviour slightly, but ultimately the same), etc. But nothing seems to work.

Same here.
I've tried the game on two computers,
one with win7pro (sp2) 64bit (i5-3350p, 16gb ram, amd radeon r7 250 1gb gddr5)
the other one with win10home (version 20H2) 64bit (i5-4210m, 8gb ram, dual card: intel hd 4600/nvidia geforce gt840m 2gb)

i even tried older versions of nGlinde, all sorts of nglide config.

I've tried windows troubleshooting, disabling DataExecutionProtection (DEP), running as admin, etc, no dice same problem all the time:

the game starts, shows the splash videos, but after I press [New game] and the intro is done, the game it crashes with
same error regarding some Splash.dll (in visual studio debugger it says access violation error).

this is what i have in event viewer:

Fault bucket -1566024447, type 1
Event Name: APPCRASH
Response: Not available
Cab Id: 0

Problem signature:
P1: HiddenEvil.exe
P2: 1.0.16.90
P3: 3810b0e6
P4: Sprint.dll
P5: 1.0.16.90
P6: 3810af03
P7: c0000005
P8: 00040572
P9:
P10:

Faulting application name: HiddenEvil.exe, version: 1.0.16.90, time stamp: 0x3810b0e6
Faulting module name: Sprint.dll, version: 1.0.16.90, time stamp: 0x3810af03
Exception code: 0xc0000005
Fault offset: 0x00040572
Faulting process id: 0x1884
Faulting application start time: 0x01d81f5b1c619709
Faulting application path: D:\games\GOG.com\Star Trek Hidden Evil\Bin\HiddenEvil.exe
Faulting module path: D:\games\GOG.com\Star Trek Hidden Evil\Bin\Sprint.dll
Report Id: ab06d7d9-6891-45ca-a59f-d9d736ca6c15
Faulting package full name:
Faulting package-relative application ID: