Messaging was still available for a long time through mbasic.facebook.com up until very recently.
They removed that feature at the time their last big messenger outage happened.
messenger.com still works on Android Firefox as long as you switch to desktop mode.
Not sure for how long though, as it looks as if they are constantly sabotaging their platform. ..and it shows: Facebook Messenger feels far less relevant than maybe ten years ago.
In my German peer group everyone now uses either WhatsApp or Signal for private and Teams+email for work related communication.
It even has some extra features, like LaTeX support.
I guess you mean it doesn't work on mobile browsers? (Never tried that)