First of all, I don't know why you're not using spoiler tags, as you're blatantly spoiling the entire twist ending super early in the thread, for all to see, barely a week after the movie's release.
As to your two main points…
1) Pop Benitez's intersex gender would ABSOLUTELY matter to someone like Pope Tedesco, along with damn near every other ultra conservative Catholic on the planet, who wouldn't be in favor of such a nomination. The entire point of the movie is that Benitez is, as evidence by his third-act speech, the only "pure" candidate in spirit, his intentions, etc. He's not coveting/gunning for the position like everyone else, he didn't make a deal with Tremblay, he's the only one in any real danger, out in the world walking the walk, risking his life, etc. Clearly, he's the best man for the job, and the movie is using his character to show how ridiculously stupid it is for the Catholic Church to discriminate based on sex, against an otherwise perfect candidate who would be ruled out because he's not technically a man. You're right, it's 100% about "subterfuge and a gotcha" because that's the only means the "underdogs" have to fight such entrenched, reductive, archaic ideas. What would you have Benitez do? Come clean to, say, Tedesco and "honestly argue his point"? Because everyone and their dog knows that wouldn't move the needle "forward." Benitez would be on a plane back to Kabul faster than you could say "laparoscopic hysterectomy," if not removed from his position altogether. All because of some dumb, arbitrary rule.
"To hide things from that body is to say I know better than this group of people." Yes! Again, that's LITERALLY the point of the movie. Lawrence, Benitez, and the former Pope CLEARLY believe that the Catholic Church needs to change. "Which, if that is how one feels, then why participate at all?" So that they can ENACT that change. So that they can help the Catholic Church become a better version of itself. That just seems… so obvious to me. Why would you not try to change something for the better - something that is so clearly in the wrong - that you care about and have devoted your life to?
2) I'm sorry, but this is a ridiculous point. In the year 2024, any organization that subjugates women, views women as less than, or doesn't allow women to lead others spiritually, etc, is a relic of bygone era. Women being equal to men is no longer a "progressive" idea, and to group it with whatever modern, cultural "wave of change" you're talking about, that the church shouldn't "bend to," is crazy. To argue that the Catholic Church somehow needs EVEN MORE time to adjust to something as basic as gender equality is laughable.
In fact, this hits especially close to home for me because someone very, very close to me, in my hometown in Texas, who was a female employee of one of the biggest (albeit non-Catholic) churches in town - one that somehow still doesn't allow women in positions of leadership - was sexually assaulted last spring by the head pastor at her church, and you sound EXACTLY like all the elders who at first tried sweeping it under the rug, then took months to take any action at all. "Taking the long view and implementing any change slowly" is what they too kept arguing in favor of, constantly putting the well-being of the church above the well-being of the victim. The pastor finally agreed to step down, but the church still didn't tell the congregation WHY he was leaving, or publicly acknowledge the victim in any way. So the victim, after giving the church every opportunity to come clean, finally decided to tell her story online. And within 48 hours it was viewed by 11,000 people, a number of which were women at the same church who then told their stories of how the "boys club" at the church often objectified them, inappropriately touched them at times, and basically treated them as lesser than. You can argue all you want that we should take our time "slowly implementing change," but when it comes to gender equality in the church, every day that we do, women are being subjugated, and such toxic, archaic thought is becoming further engrained, emboldening the worst of those in positions of power to do exactly what they did to my sister.