Frameworks like Laravel make this easy because they allow you choose different backends for your DB, queue, and cache, among other things, all or some of which can be an RDBMS like Postgres or MySQL/MariaDB.
When your app's need outscale what the RDBMS can rationally handle, then you can dedicate the resources to switch to a more specific tool as your growth has indicated you should.
I can't say what the percentage of apps that can handle things this way, but the anecdotes mentioned in this thread are clearly the cases where growth outstripped what a traditional RDBMS like Postgres are well-suited for, and so migrating was the right thing. The probably was likely identifying the need or delaying the transition until it was too painful. I don't think there's a magic bullet here, but I also don't think that starting with a dedicated queuing store tool is also always the right thing to do.