I've been commuting since the mid-80's and started using clipless pedals in the mid-90's, I've even done several loaded tours thru the Appalachian mountains with them. I don't think they helped me that much climbing those steep mountains.
What I like about clipless pedals, is that I'm somewhat of a speed demon, probably because cycling is my primary form of transportation and I need to NOT spend so much time commuting. Also, since I ride in traffic a lot, there are times I need to do a quick sprint, based on road conditions and traffic. If I were to do this on flat pedals, my feet would constantly be slipping off -- I know this from experience. I realized this early on and started using the old-style cages, but eventually switched to SPD's and will never go back.
As for improving climbing, the best thing you can do is to start lifting weights. That's just not my opinion, there have been countless studies showing weight training improves cycling, despite many cyclist being dead-set against the idea of lifting.
P.S. The very first loaded tour I did thru the Appalachian mountains I did in the old-style cages and all others using clipless pedals. I don't recall any difference in performance of the pedals. However, I very much remember the pain of pushing those pedals up them steep hills.