We are always learning along a continuum - in any dojo there will be people who cannot throw me and people I cannot throw. So yeah, with a newbie, I actually keep my idealized ukemi a step ahead of their nagewaza and not only fall for them, but guide them through my body. Past that, my willingness to allow myself to be moved is based on starting the first time as uke with total cooperation, attacking to their center and then essentially allowing nage to lead as long as she has found some kind of connection to me (the kind being related to a sense of her skill level) and letting myself follow where her leading is taking me. If she disconnects I let her know and let her refind it rather than start over. With peers and seniors of course I expect more and don't simply follow nor do I resist statically if they mess up, but gently ramp up my own attack on their center to give feedback that can be used to self-correct.

I don't see my interactions at the dojo as "winning."

FWIW, there were a couple of dojos I was a member of that were not paragons of cooperation. They also were nothing about what you would call "aiki" - they were purely into the mechanics of imposing the asked for technique with lots of muscle power regardless of the form of uke's attack. So I don't have a very high opinion of non-cooperative training.
However my view is that within cooperation there can be, and should be, the contract between partners to test one's limits.

I believe it is Chuck Clark whose teaching is based on the principle that one should succeed 90% of the time in order to learn. I think this is true, whether the partner practice is slow and aiming for feeling connection or faster and working on the form of a specific technique - it is in the 10% failure that one learns what is needed in order to progress and in the 90% success that one starts building the incremental muscle memory (for lack of a better term) that drills the skill in.

I think part of the problem may be semantic. Bear w/ me for a moment on this: I have felt/seen two very different kinds of aikido, as have many of us: the "wow that was a strong throw that sent me across the dojo" and the "wow how the heck did I end up here". I think for many of us the goal is the more elusive, harder to find latter feeling.

So when you write of "center driven power to a level where few can stop you?" I have a feeling that however you may mean it, for many on Aikiweb this smacks of the style of aikido they have experienced (as I did in dojos like the couple I used to train at a long time ago) and are not interested in doing.