I would probably do it for home, maybe. But creating a health check from the app I already am building and pinging that does the job and isn’t too hard. In addition that tests the secrets provided to the application. Secrets management and rotation is also a challenge!
But I can see why some people might want to ping their DB.
At work we also monitor the other way for example alert if CPU or Memory usage on the DB is high. But this doesn’t replace a ping check just augments it.