On iOS with Opera GX or Firefox (all up-to-date, has been going on for the ~year I've known about chat), whenever I click on the textbox on the desktop chat view the phone zooms in on a small corner of the textbox. If I zoom out and click on it again, it zooms back in.
If not for this bug, I would likely use desktop mode all the time so that I could access all of chat's features, but it's annoying enough to not bother and only switch back and forth when I need to.
I don't think we can even control this behavior -- how phone browsers deal with non-responsive pages is usually up to the browser.
For what it's worth, I'm not seeing this issue in Firefox on Android. But as far as I know, Apple only recently announced they would allow alternative browser engines, so presumably FF iOS is still using a wrapper around Webkit, and thus isn't a "real" Firefox? The same probably goes for your other browser, which means it's not surprising that they exhibit similar behaviors.
In either case, I don't think hacking around the existing desktop view to fix issues in phone browsers is the way to address the limits of mobile chat. Rather, as you're pointing out in your other post, the live chat view should either actually be responsive, or the mobile-optimized view should be brought up to par.
