I'd say the answer to this question is yes.
Being ambidextrous while climbing will make you a better climber, because there are certainly routes that have forced, right-handed or left-handed cruxes that simply can't be done the other way around.
I've seen people really struggle on problems that force them to rely on their weaker hand, instead of committing to going for the next hold with the hand that makes sense, they'll spend excessive amounts of energy trying to match hands on a tiny geek so they can attempt an awkward off-balance cross to the next hold with their strong hand, which of course doesn't work much of the time and they end up taking a fall.
These types of climbers are typically found in the gym, where they constantly set problems that suit their style, and their strong hand. I knew one guy that seemed to always make the crux of his problems to be a big move off of a strong right hand pinch, so I'd set problems with strong left hand pinches just to throw him off his game.
Climbers naturally have a dominant hand, but if you asked me which of my hands was stronger I'd probably have to say both hands were equally as strong, but my right hand is more accurate. I don't think I can hold on with one hand longer than the other, but I certainly have better aim with my right when I'm making big dynamic moves. If I did have a stronger hand, it would be my left, because it's the one that's hanging on most of the time when I'm aiming for a dead-point dyno to a tiny crimp with my right hand.
I think you could get away with being a climber that is dominantly right or left handed, but it will put a limit on your potential. I guarantee that the strongest climbers are indeed the more ambidextrous ones.