If I go to /settings/panel/home/ on mobile (Android + Chrome), I can't rearrange the pinned applications (or possibly I just can't figure out how to).
Yeah, not much actually works on mobile, it's current level of "support" is read and comment on most things only. It's not expected to have full functionality - like any drag and drop interface - at this time.
This also isn't accessible. I'm not sure what the best approach is in general -- we could try making just the "grip" part draggable on mobile as a first approximation and see how that goes. I think Workboards will eventually need a "Task > Move To Column..." action since I don't see a reasonable way to do it in the UI for mobile/assistive.
fwiw in Trello boards you can drag tasks in workboards pressing a task and dragging it. I won't bet on the accessibility of this approach, but our Trello users seem to be very happy about it. Apparently it's a relatively frequent task on mobile, just like other non-typing-a-lot tasks.