Would projects such as Ungoogled Chromium be affected? Or possibly, is it something one could cut out from the source code?

I think it would be risky, companies will roll it out think people who are on chromium will move to chrome, or their browsers will support this. If people move to Firefox, companies know that a percentage of their users will be prevented from using this, and it could cost their marketshare/revenue. Google cannot be trusted to dictate web standards any more, and Mozilla is the best placed to break that hegemony.