The evidence to the contrary came in the form of power-meter data from pro cyclists. It shows that professional cyclists instead use their power only in the down stroke of the peddle and have a far less “even” power through the cycle. Their technique is much more similar to a piston engine than a rotary.

Doesn't "spinning out" mean your power output is bottlenecked because you can't move your legs fast enough? It only happens when your gear is too low. I don't see how that's possible while climbing, regardless of what pedals you use. I ride with flats, and there's a 10% incline hill not far from where I live. I've climbed it several times (at about walking pace) and never had any problem.

Alternating left-right pushes feels very different than trying to maintain a continuous circular motion with quasi-equal force throughout. But unscientifically I can't notice any difference.

Clipping in is mostly a cargo-cult among non-pro riders

Otherwise there is no benefit and a lot of downsides in terms of how likely you are to get injured by poor cleat placement or saddle position.