Everything worked fine back on Windows 3.1. Then you upgraded to Windows 95 and now your favourite game crashes. Why would you not blame Microsoft for this?
Browsers and engines are also regularly patched to improve web compat. I remember V8 "recently" released an update to improve the performance of a very common module. The details escape me however.
GPUs and games have a vested commercial interest in ensuring they work.
On the other hand, both the website and browser are mostly provided for free. Paid websites often do put the extra effort in to make it work everywhere, eg IE/Safari compat.
Someone’s paying for you to use the browser, even if it’s not you - it’s not like Mozilla is hoping everyone will stop using Firefox. So browsers do have compatibility checks for websites sometimes.
