I just tried, and my macOS up todateFirefox (still Sonoma few weeks), doesn't work. Nor does Waterfox (Firefox derivativ) that I've been using more lately. Could it be some setting I need to set before it works?

Unless I misunderstood your question, are you simply trying to use emacs bindings in all text areas of the browser like the URL bar, any text fields on webpages, being able to move to the beginning or the end of a line, backspace, etc., go to the next or previous line, all with emacs bindings? That has worked for decades on Mac and it works for me now in all browsers.

Idk, its never stopped working for me for the past 10+ years. And Firefox has additionally always supported ctrl-n and ctrl-p working in the address bar for suggestions while Chrome has never supported this

Bit late to reply, but seems ^A and ^E (jumping line begin and end) at least do work in Firefox.

But for example ^T (letter transpose) do not work on macOS Firefox, which I'm quite accustomed to use also. But ^T works fine same macOS Safari and Chromium.

Firefox apparently has its own input method which do not implement all what macOS supports.