tsh in Connect terminal stops working after logging out of a cluster

[ravicious]

Not attaching this to the test plan since this probably never worked. It's present on v16.0.0 as well.

Repro:

1. Connect to a cluster.
2. Connect to another cluster and log out of it.
3. Open a terminal tab in the first cluster and execute tsh status.

Both tsh and tctl think you're logged out because $TELEPORT_HOME/current-profile points at a profile that has been removed.

The notion of current profile as understood by tsh has been largely inconsequential in Connect. Switching a profile in the top right doesn't change the current profile in tsh. Perhaps it should, but there's the problem of functions such as lib/client.RetryWithRelogin changing the current profile as well (#25806 (comment), that's been the case in April 2024, idk if it's still true).

Perhaps Connect should sync its current profile with tsh's current profile, especially once we get around to #25806.

Until then, a good enough solution would be to pick the first cluster during the Logout RPC (or Remove Cluster RPC? We still haven't unified them) and set it as the current cluster, if any.