there is a lot of output on the neck PU with the E-string but you probably wouldn't notice because many players are not used to hear the fundamental of an "E" but its next harmonics. The so-called 'solid underlying bass' is exactly that what you can hear/feel if only the rig is capable to make it hearable (or give enough pressure onto the speakers).
Exactly. I was just in the studio, where the monitors were well equipped to produce very low frequencies audibly (much unlike many bass cabinets!!). I only used my neck pickup to record all of the tracks and the bass was as loud and "in the mix" as ever; yet I have often thought that pickup was quiet when running it through a bass cab. I'd like a bass cab that could produce lows as well as a studio monitor.