In Firefox menu bar click Help then Troubleshooting Information. It will open a new tab with info. There is a section called Graphics. You may find error messages in that section. If so that would likely be caused by the GPU driver not being good enough.
The newest version of Firefox has very detailed Graphics info so I guess hardware acceleration plays a bigger and bigger role.
You mention Firefox is a mess and if that's the case you should try starting it with add-ons disabled. Menu Bar > Help > Restart with Add-ons disabled
The reason some browsers may work fine despite faulty (less than ideal) GPU driver is that they have fallback options such as using the CPU if GPU hardware acceleration is unavailable.
Despite my GPU driver being faulty according to Firefox, Firefox did still work on Youtube. It was my preferred browser Pale Moon that for some reason didn't fallback to CPU rendering when GPU acceleration didn't work.
Worth to mention is that I had no other issues whatsoever. Pale Moon didn't work on Youtube and the driver was marked as non-working in both Firefox and Pale Moon, that was it.
Since I've used Firefox since version 3 and Pale Moon since Firefox switched to Australis UI (Firefox 29), I don't feel like using another browser even if that means Windows 10 instead of Windows 7. Chrome lacks the UI customization I'm used to so I only use it as a reference for bad sites that don't conform to web standards, but Chrome standards. Thankfully very few sites are Chrome only.