The models that adjust for the potential confounder reduce variance of the treatment effect estimator ----getting increased power. In general that's what you want. We want to avoid the mischief of Type 1 error but you didn't find that to be different for the models you tested-------so
I'm not sure why you seem to come down for not adjusting in your conclusion.

I do favor adjusting, though I think using the two step process is more principled, yet also has the efficiency benefits. Hopefully, I'll get to show that sometime in the future.