Add health probe for database

Context

The container registry has a /debug/health API endpoint that can be used for health checking the registry and its dependencies. The functionality is documented here.

Problem

We have no probe for the database dependency. We definitely should have one given the hard dependency that it represents for the service.

This was identified in !1628 (comment 1927787782).

Solution

Add a database probe and expose it in the /debug/health. Make sure this is integrated/adopted for GitLab.com (can be a followup).