There’s this video around that puts people in a VR helmet playing A/V with a delay, even the lowest setting completely destroys any form of efficiency.
Anything below 60Hz in VR has a significantly increased chance of giving motion sickness due to latency.
Playing Super Mario Bros in an emulator is strangely subtly harder than on the real NES hardware, just because of a slight lag added (can’t recal on NES but on SNES buttons literally bit-flip the RAM, hard to beat that). You can train all you want on an emulator, when you get back to the real thing it’s obvious it was just so crippling.
Try Guitar Hero / Rock Band with various audio / video latency settings. 10ms is obviously impactful, anything higher borderlines on unplayable, and even 5ms has a subtle impact increasing missing notes.
As for pro musicians, they absolutely loathe latency. This is why bassists (tempo) / drummers (rhythm) / conductors (tempo) exist: they serve as a single source of truth to sync on. In fact, we’re so sensitive to latency that the slightest deviation is unbearable, hence why following them put you back in sync. Same for effect pedals, amps, feedbacks... any latency is destroying.
Try playing on Zoom, it’s absolutely impossible (unless you assume latency is fixed, which it is not, and play in a tempo matching latency exactly, and follow the lead but time-warp your play by one beat, which I imagine is possible as a sort of canon but absolutely horrendous) unless you completely ignore the other part and find a common time source to sync upon.
Now you mention Quake 3, and it might just be you were severely misled by several critical innovations that game introduced, like network prediction and frame interpolation, negating most of the practical effects of latency.