This is the classic example of why you should always set the foreground color when changing the background color. Otherwise you end up with <code> blocks that are black-on-black. It probably works fine if your browser is in dark mode.
A workaround is to open one of the code elements in the browser's debugger and add color: white to the :not(pre) > code style.
Does it allow truly persistent mosh connections? I can't remember which iOS client it was, but I've used one which forgot I was connected if I left the app for long enough. The mosh process was still running on the server, but my iPad had forgotten how to talk to it.
I believe the primary use-case for devices like this is debugging "Why isn't this server rebooting?" without driving to the datacenter. Good luck figuring that out with AnyDesk or SSH.
A workaround is to open one of the code elements in the browser's debugger and add color: white to the :not(pre) > code style.