In the 1960s and even into the mid-1970s those of us who were airline pilots based at LAX could fly light singles into LAX and park on our airline's maintenance ramp (not the passenger terminal ramp).
(Why) did it stop?

Well, this may not be the direct reason but it may indirect reason...

On August 31, 1986, a Piper Archer mid-air collided with Aeromexico
Flight 498 DC-9 in approch to LAX. 64 in DC-9, 3 in Piper and 15 on
the ground died. That event caused FAA established Class-B airspace.
This greatly reduced the small GA airplanes accessing the major
airports such as LAX.

You can read the full story at AOPA:
http://www.aopa.org/asf/asfarticles/2001/sp0101.html