I am experiencing something similar. Windows 10.
I downloaded the game from here https://files.ioquake3.org/Windows.zip, unpacked it then moved the original game .pk3 files to baseq3 of the unpacked folder.
I also tried following the instructions in https: //ioquake3 .org/help/players-guide/#windows (running installer, then replacing files), but the result is the same.
Notably, when I launch ioquake3.x86.exe from the old installer instead of ioquake3.x86_64.exe from Windows.zip, it runs smooth, not as atrociously as ioquake3.x86_64.exe.
My original Quake III Arena from steam also runs smooth.
My GPU is an old-ish (~2015) Nvidia one.
This looks very similar to Engine runs at less than 1 FPS on Windows XP.
And just as I was about to press “post”, I launched the game once again, and it worked perfectly! Both the “installed” version and the simply unpacked Windows.zip.
Frankly, I have no idea what fixed it.