LAX, an air traffic control tower, does not employ ERAM. Los Angeles Center (ZLA), the overlying air route traffic control center, does.

Why do secretive spy planes need to file flight plans?

Because they're not always doing secretive stuff, and even their secretive missions may start and end in airspace they have to share with friendly airliners.

Sometimes it's a deliberate "we see you", too. https://twitter.com/Gerjon_/status/1475444704250630148

That was a training flight, it's in the PDF.

I imagine operational flights might "officially" start and end as routine training flights, at least on paper.