An aggregator whose pooled credentials are all exhausted/dead still counted as
an authenticated provider during no-provider /model resolution. It then won the
model-name match, was set as the sticky session provider, and poisoned every
later switch with "empty API key" errors while still routing through the dead
aggregator.
list_authenticated_providers now requires a pool to have at least one available
entry (has_available, not has_credentials / bare key presence) at all three
credential-pool gates. Simple token-style entries that don't parse into
exhaustion-tracked entries keep the prior behaviour, so providers whose creds
live only in the auth-store credential_pool still appear.
Fixes#45759
