The bottom line is Firefox could support whatever mechanism is used in Chrome, but despite getting $600 million a year from Google, they prefer to spend it on diversity nonsense rather than a working browser.
The Mozilla Corporation reinvests all profits into its development projects, most notably Firefox and Thunderbird. Its non-profit parent, the Mozilla Foundation, can do other things as well, but its mission is about the health and openness of the internet, data privacy, integrity in AI, and holding tech companies accountable. I can't find any direct donations other than $10,000 to OpenSSH. The only mention of diversity I could find is in their hiring practices - that doesn't cost anything.
Firefox is an excellent browser - if you think it doesn't work, you haven't used it. One of the reasons it's great is that it is not running the Chromium engine (what you called "whatever mechanism is used in Chrome") like Chrome, Edge, Opera, etc. Instead, Mozilla continues to develop and maintain its own engine in order to stay independent and focus on privacy and extensibility. That is a good thing.