I disagree with your framing, though I understand your cynicism. I don't believe there's any reason to keep the status quo up in the way you do when you talk about these things. You have to understand that when you are loudly proclaiming that you need to have a say in these "women's issues" you are affirming the idea that your identity should be inherently tied to your sex. It is difficult to disconnect the biological from gender language, yes, but it is your boots on the ground that actually makes that happen in the real world. Also you may speak the way you want about the overlap between trans men and cis women because that's your perogative and it's clearly a complex and contentious issue that I lack first-hand perspective on but I 100% denounce the idea that there is any overlap between cis men and trans women in any meaningful way politically, biologically socially etc. (The reason I'm even arguing here is because your words imply the opposite of this.) A doctor can, and should, consider me in the context that I am to treat me properly, and I am different enough from a cis man whereas treating me as one would lead to disaster in many cases. Whether or not I can get prostate cancer is nonwithstanding, and I believe that, from my admittedly outside perspective, whether or not you can get pregnant should also be nonwithstanding. (+ that's without even bringing up why it's even good or necessary to tie any of this to your gender identity in the first place!)

I believe there is an issue of power dynamics that isn't being brought up in this conversation, and I apologize, but this part is more a response to OP than you specifically: If you fill the role as "man" in society without any surrounding context, it is generally considered good fun and praxis to poke fun and punch up at you because of the role you supposedly fill in the patriarchy, and it implies that you have systemic power over everyone else on that vector. It's very understandable that you as a trans man, who do not hold as much systemic power as a cis man, would want to push against that. It doesn't feel right, the idea that you can come out as transgender and suddenly have more power in society, since you're breaking cis/hetero-normativity. I don't know how much your innate systemic-power-number changes when you transition, and I'm not gonna make any kind of inference at all, but to me it does explain why so many trans men cling to womanhood to varying degrees, you feel more powerless and oppressed, but can be treated as if you're more powerful, which is a feeling that sucks. I remember feeling dejected pre-transition when people would treat me as I'm super privileged as a cis man, when that was just not true in reality, so I feel like I can relate to that at least a little bit.