There are two parts to this. You are correct that RDS doesn't help you with picking the index strategy, or optimizing queries. I don't see that as running the DB though, that is how you interact with it once its running. What it does do it help you reliably run the DB server itself.
Without any effort you can stand up a redundant, high availability deployment. With all of the data encrypted at rest. And configure nightly backups, which are stored on redundant storage in multiple physical locations and also encrypted. You can then restore those backups into a working system with the click of a button. Oh, and minor version patches happen automatically with no downtime. And you can click a button to do major version updates.
The last time I did analysis on it, which was a while ago, all of those features cost us less than 8 hours of my time each year. It would probably take more than 8 hours of my time each year just to handle security patches on the systems. Let alone the amount of engineering that it would take to get a system as redundant and reliable as a DB in RDS. I will happily pay them to take all of that off my plate so I can focus on other things, like optimizing the queries.