I can search millions of records in a split second in SQL. But searching records on my hard disk takes enormously more time. The solutions out there for speeding it up (like Windows Indexing service) rely on asynchronous indexing that's patched on top of the filesystem instead of integrated realtime within it, and are subsequently brittle (in my experience), slow down your computer at inconvenient times, and too often aren't instantly up date.
Being able to have stored procedures execute when changes trigger would be another interesting aspect that could take the place of event filters and file watchers (especially if critical portions could be configured to happen sequentially & atomically, akin to IRQ handlers).
I feel like databases have gotten lots of love over the decades and are super optimized (and by this point are pretty dang reliable). Filesystems feel like they lost their time in the spotlight and have floundered with less innovation.
Hardware that calculates CRC's of blocks in realtime would also be awesome (imagine what it would do for sync). The ironic thing is your HDD already does this, there's just no way to tap into that information at the software / filesystem driver level.