IIRC changing to the Classic theme disables the Desktop Window Manager (DWM). Basically I think it reverts to an older and significantly different compositing strategy. I think it disables GPU acceleration of desktop windows so it probably reduces overall system performance. Apparently the option is also removed in Windows 10+.
It's surprising if the DWM has poorer v-sync accuracy than not using it... Recently I saw
about presentation models in Windows 10, and it turns out it's super complicated with several different presentation modes with different options and latency/accuracy trade-offs. I guess on some systems, different composition strategies perform better or worse, and there's probably no single one that works best everywhere. Still, this suggests any v-sync timing issues are coming from somewhere in Windows itself, which is kind of surprising, usually the OS internals are really well engineered
FWIW, on every modern Win10 device I've tested, Chrome/Firefox/Edge can v-sync to within ~0.1ms. Maybe they've only optimised for accuracy on modern hardware.