You're better off at separating headphones and binaural sound processing part.
This is what gaming branded Chinese garbage has for frequency response:
https://uk.hardware.info/reviews/60...gher-quality-testresults---frequency-response
Roller coaster's loop wouldn't look out of place in those curves...
So those will make garbage out of good signal anyway.
And likely bundled USB dongles etc are rather bad to start with.
But first of all is your environment noisy or quiet?
Noisy environment needs closed headphones.
But if there's no need for sound insulation open headphones are better approach.
Those have wider sound stage (good for binaural sound) and if you happen to suffer from high temperatures at summer they don't increase sweating.
Anyway you woud likely want some strength in bass over neutral.
For competitive gaming neutral bass is best, because it keeps foot step sounds etc sound clearer in relation to explosions and such.
But that comes at expense of "fun factor" with neutral bass easily feeling very "shy" and empty.
Meaning sound is always compromise to one way or another and balancing between things.
So am I right in thinking that some strength in bass is wanted?
In open cans Beyerdynamic's DT990 Pros are very good for their price.
They have wide sound stage and for open headphones good amount of bass, without it walking over everything else like typical strong bass gaming headphones have it.
(DT770 would be good closed cans for that price level)
While AKG's 701/702 with neutral bss are one of the best headphones for that competitive gaming, they suffer in that fun factor.
K712 is AKG's headphone meant for balancing between neutrality and some strength in bass similar to DT990.
Then part of getting that good binaural signal for 3D sound with headphones needs to be included.
Stereo is obviously limited just to "left and right" direction and 5.1/surround sound just can't be downmixed to two channels.
That's the area where sound cards have clear advantage over integrated.
Windows 10 also has now some kind headphone sound mode, just can't find videos for checking how well it works.
For sound cards finding comparison is easy.