“That Paul appeals to elsewhere” - you’re thinking of 1 Timothy 2:14?
Even if you’ve got another passage in mind, these two passage 1 Cor. 11: 3 and 1 Timothy 2:14 are - as you note - not embraced by our side. When they come up for exposition, our side almost always (1) slides over them as if they weren’t really there, or (2) turns exegetical and/or expositional cartwheels in order to explain why they do not seem to mean what they obviously require any sane reader to see in Paul’s statements.
Why does this happen?
Well, in 1 Corinthians 11:3, Paul tells us that Christ is the head of every man. And, the woman? Isn’t her head Christ too? Evidently not, as Paul immediately goes on to say that her head is the man. Taken together, these ideas - every man has Christ as his head, while woman has (some) man as her head . . . well, we all know how that idea fares in this fourth-wave feminist climate!
Moving on to 1 Timothy 2:14 - well, dang it! Paul says Eve fell into transgression because she was deceived, and for that reason all women in the church are by Apostolic prohibition constrained from teaching men.
Why that constraint? Is it a punishment? How unfair to penalize all those sisters with the spiritual gift of teaching!
Or does the prohibition look to her being deceived?
Religious feminist: “You’re not saying that women are by nature more deceivable than men, are you?”
Complementarian: "Oh no, no, n!. Paul didn’t mean that!
Religious feminist: "Well, then, why does Paul fault Eve for being deceived? What’s that got to do with anything anyway?
I’ve seen this dialog play out ad nauseum for decades now.
So, yes - our side has some work to do, and that work needs to be different from what complementarians have done since the Seventies (i.e. to retreat, shilly-shally, and so forth). Our side needs expound the NT’s teaching on the nature of the sexes (here’s where an exposition of the glories in 1 Cor 11 is ground zero) and also the relationship between the sexes in marriage, family, church, and society (and there are differences of kind and degree, depending on the stage where the sexes are relating to one another).