Is there a performance benefit to be had by managing the memory and paging yourself? Yes. But eventually you will also consider running processes next to your database, for logging, auditing, ingesting data, running backups, etc. Virtual memory across the whole system helps with that, especially if other people will be using your database in ways you can't predict. As for the efficiency of MMUs and the OS, seems like for almost all cases it's "satisfactory" enough[1].
[0] http://denninginstitute.com/pjd/PUBS/bvm.pdf
[1] From 1969! https://dl.acm.org/doi/pdf/10.1145/363626.363629
The reality is there will always be a hierarchy for storage, and paging will always be the best mechanism to deal with it. Because primary memory will always be most expensive, no matter what technology it's based on. There will always be something slower, cheaper, and denser that will be used for secondary storage. There will always be cheaper storage. And its capacity will exceed primary, and it will always be most efficient to reference secondary storage in chunks - pages - and not at individual byte addresses.