You could add more monitoring checks, than e.g. checking if the site is up and its content serves a specific string (I do that with Icinga on monitoring-portal.org).
If you for example enable the REST API, there’s many things you can test/check. Even insights into the applications currently running. Since Discourse runs in Docker, there’s a variety of APIs and checks around to go further with Redis, PostgreSQL, Nginx, Ruby-on-Rails, and so on.
You may also go the route with full application and UX monitoring with end2end tests. There’s certain frameworks around like casperJS, not sure though if that works with Discourse. Probably better to have such things automated via API.
But still, you could go the route to trigger events and expect something in return on the current site.
I don’t do that yet, but it is on my list to increase monitoring and always know if the site is fully operational, or if I have hit a bug or a regression.