The complication reside in the definition of the health status (and its usefulness).
You mentioned the definition being
service is up and web app application is operational and for that indeed checking the homepage should suffice.
However that doesn't mean the service is properly configured for and correctly performing all the jobs that it's supposed to be doing. Without a consistent, universally acceptable definition of a health status there is no point of providing such information as a single, one-stop location to check.
But Jenkins offers via its Remote access API various checking points that users can combine to derive their interpretation of the overall service health status. For example maybe checking the status of all the configured jobs or at least that of the critical ones would be more useful, which could be done using the
The JenkinsAPI could also be of interest (better docs at least), found via Where can I find jenkins restful api reference?