Having created a fully functional SQL Server Always On Availability Group, I connect using SSMS.
Upon trying to expand the database, I get this message: "The database [...] is not accessible (Object Explorer).
The stack trace mentions get_CanGetChildren().
File this one under "sometimes, our lives are meant to serve as a warning to others."
I was connecting to the Windows Server Failover Cluster, rather than the SQL Server listener. SSMS / SQL Server is happy to let you do this, but then you're not able to expand the database.
When I connected via the listener, it worked without issue.