>Either the methodology of Cox Regression is valid or it isn't regardless of the amount of change in p value.<
In theory, yes; in practice, no.
If I were a reviewer at the FDA or another regulatory body, I would tell any company that comes to me with a raw p-value of 0.33 and a Cox-adjusted p-value of 0.02 to redo the trial and come back with clean data. Even if it’s a supporting trial. (Or a pivotal trial masquerading as a “supporting” one :-) .)
In other words, I stand by the fine-tuning radio knob as the proper way to think about Cox from a practical standpoint.