I just signed up with Aiven myself to test it for myself, and I already see the problem.
Our servers allows outbound MySQL database connections on the normal MySQL port. Access to custom ports is blocked on our hosting.
But Aiven appears to host their MySQL databases on a randomized port number. And you’re basically guaranteed that this random port is blocked.
So yes, you should be able to connect to external databases, just not in the way that Aiven has setup their databases.