I think SSH tunnels are exactly the type of thing that should be handled by a different tool. It's a lot of complexity to add to the app, when the app (and any DB client) inherently already supports connecting via an externally managed tunnel. An established tunnel managing program, or just your terminal, would be better at managing tunnels than a tunnel tool tacked onto this app, and the interoperability is already free.

Tunnels have so many configuration options that you'd either have to support raw configuration in the app (at that point just DIY a tunnel in the terminal), implement something incomplete, or build a full featured SSH tunnel manager into the app.

SSH tunnels can be totally transparent to DB clients. Unlike DB application-layer stuff like credentials, where the DB client needs knowledge of them, I don't see the cost/benefit of integrating SSH tunnels into the app itself.