Re: Taking out a pickup.
Yes inasmuchas the instrument would have a vastly reduced range of possible sounds.
As described earlier, a lone J pickup, close to the bridge, will emit a bright, wiry, almost piano-like sound.
A lone P type pickup will emit a stronger signal with far more of the fundamental frequency information but, possibly, not as much high frequency detail as one might wish.
The idea behind having two pickups is to be able to blend the best qualities of each type in its respective position to arrive at a satisfying sound for whatever music style you happen to be playing.
To my way of thinking, two not especially good pickups is still better than just one not particularly good pickup.