For the vast majority of projects using Linux as their server OS, wouldn't it be simpler to just have systemd start the service and include `Restart=on-failure` in the unit file?

I don't know about vast majority, but for many, yes. Where health checks come in handy for me is when my services are unreliable. I find it pretty common that my services don't crash outright, but they get into weird states that make them nonfunctional. systemd doesn't help you there, since it only cares if the process is still running.

Something like stealthcheck makes it easy to run smoke tests (or even full integration tests) and force restart the service if they fail.

Then again maybe I'm just bad at writing services.

The problem is when the host goes down. Most kubernetes installations will handle that. Is it worth the administration overhead? Perhaps for a large company.

For our production deployment we have 2 instances of stealthcheck on separate EC2 nodes. AWS takes care of keeping the nodes up and stealthcheck takes care of the rest, in 10 lines of config.