It is.common to have an anemic j in a lower end PJ (and even USA made, these were lower end instruments), but sounding like am radio is a bit extreme, could be just a crappy pickup or a bad pot. A decent jazz pickup should still have useable low end unless the pickup is super close to the bridge, but it shouldn't sound like an am radio. I know when I got my first jazz bass, the bridge pickup was way fatter than the jazz pickup on my PJ ibanez TR.
A better pickup is likely in order, or at least check the pot and wiring first. Perhaps one of the gurus can chime in about what j pickup was in the early forums. A cheap P pickup in the P position can still sound fat and modern compared to a cheap jazz pickup. Even a fender TeX Mex mim jazz pickup would be a big improvement over some of the Asian pickups that peavey was known to stick in some basses. Or a.dimarzio or Duncan or fender USA pickup. If your forum has a USA peavey p pickup that will probably hang well with an upgraded jazz, but if it is the thin wire Korean pickup, you might want to change that one too.
My foundation bridge pickup is definitely less low endy than the mid pickup, but am radio is not the term I would use. It is closer to the bridge than even a 70s jazz, Lemmy is the way I would describe it, and it does not kill the low end while blending, but solo'd it is not as thick in the lows as a good fender jazz pickup in the 60s position. It is my one gripe with the foundation, but I have been waiting for a chance to try it in a punk situation or a motorhead cover
