According to ICC color correction in Firefox, Firefox has been capable of reading colour profiles in images since Firefox version 3 (released 17 June, 2008). This was configurable, and initially defaulted to 'off', but this default behaviour was changed to 'on' in Firefox 3.5 (released 30 June, 2009).

However, this comes with a big caveat. It seems ICC profiles come in two versions, and Firefox only reads the older version by default. Quoting Wikipedia, "Two main generations are used: the legacy ICCv2 and the December 2001 ICCv4. Since late 2010, the current version of the format specification (ICC.1) is 4.3."

There is a configuration setting to get Firefox to read ICCv4 profiles. To enable this, enter about:config in the navigation bar, and search for and change the setting gfx.color_management.enablev4.

As mentioned by the OP, this is all well and good for your own browsing experience, but it's quite unlikely that other Firefox users will have gone and fixed this configuration themselves. It looks like there is some ongoing work to address this by Firefox developers, but since the linked bug is 13 years old and still open, I wouldn't hold your breath. If you want to fix this yourself for all Firefox users, here's how to contribute to the Firefox code base: https://developer.mozilla.org/en-US/docs/Mozilla/Developer_guide/Introduction