Also, I hardly see gender critical people be any more accepting of the "years on Hormones, multiple rounds of FFS, post-buttom-surgery, unclockable-if-they-dont-tell-you"-level trans woman compared to the "day-one, pre-everything still trying to get rid of the beard stubble"-level trans woman.

there are plenty of trans women who are not "pre everything" but simply do not want surgery. It's hard to find clear statistics but even among those who get any kind of medical intervention, it's a very small percentage who have any intention of having their genitals removed.

But the key thing here is how much it ends up affecting people's lives - I'm not religious but don't much mind if people say things like "god has a plan" or "I'm praying for you" - I take it as well-intentioned and translate it in my head to something along the lines of "I want things to go well for you". So I thank them for the intent and don't argue over the details. But if they start claiming that I am scared of God and just need to submit, that various words have specific new definitions (like the lingo of cults), that I have to do the sign of the cross every time we interact, it's going to become more unwieldy. Trans women who have removed their genitals, mostly pass, and usually don't ask to be included in private women's groups are like polite christians who might say "god loves you" every now and then but don't push it. Most people will just smile and nod. But even the spiritually open-minded can get pretty pissed off by a new religion who is trying to change the law so everyone has to follow their belief system. Trans women who want to change the rules of sports, of privacy, of language, for no reason except their own belief system are imposing their religion on the rest of us.

I don't think any transgender people are really claiming they never had the body they struggled with.

Then why can't they accept being "trans women" and not demand to be included in the category "women" in every case? They are not women, they are men who have struggled with and cosmetically altered the body they have. They are a different set.

They might try to keep it a secret if they are stealth, but that is not the same.

If they want to be in the closet about it, that's their thing to deal with in their own lives. I wouldn't think it the best way to go, and I can't imagine it can work in all relationships (maybe superficial acquaintances, but not real friendships, and your family always knows). What matters is that they don't claim that they are actually women and should be allowed to e.g., compete in women's events - just that they are not "out" about being trans women, and therefore, have to make excuses about why they can't compete in the women's event.

No, I don't... because, at the end of the day, part of the transgender identity is also the desire to be socially accepted as part of the same group as the cisgender people of the respective gender.

So, although there is no definition that includes both, their desire to be included in the definition should change the meaning of our words? Should we change all words to work that way? A lawyer is a person who has finished a law degree or anyone who wishes they had a law degree? Your husband is the man you married, or any man who wishes they had married you? Why is the desire of the trans woman to be included in the word more important than an objective definition?

How would you like it, if every instance of the word "woman" would be replaced with the word "ovary-haver"?

No, that is what you are trying to do. The word woman already includes the idea of having ovaries, along with other biological realities. It also includes the idea of belonging to the human race and being a person and having the experiences that come with being biologically female. If I was going to ask about something specific to ovaries, I might want to also want to include people who used to have ovaries, but I would not care to include men who have no idea what having a female reproductive system is like. So, the word "woman" is perfectly good. I could also use that word to ask people about the experience of growing up female, or being worried about being pregnant, or menstrual cycles, or breast cancer or any number of other things. But what would you need the category of "woman" for to address "cis women and trans women" ? What experiences and commonalities do they have ?

It would mean, that every use of the words "woman" and "man" would be reducing the person in question to their gonads, with no regards to their identity.

Woman and man aren't the entirety of anyone's being. You also have a whole lot of other words that you can use, plus your own name. Your "identity" is not ensconced entirely in the concept of woman. It just means you are a female person. Don't get so stuck on that.