I also use FF on daily basis and I don't remember having such problems. Some webapps like Skype were blocking FF user-agent on purpose (changing User-Agent made it work). Maybe I use different sites than the one complaining but it may also be worth checking addons and considering starting FF in Safe Mode (which disables addons temporarily). Sometimes addons can mess up sites in unexpected ways.
I find this really interesting, and would like to hear more. I've been a Firefox ESR user for almost a decade, and apart from some breakage on some videoconferencing sites, and sometimes having to disable uBlock Origin, I've not faced an unusable site. And this is on ESR!