This is, unfortunately, a very unkind way to interface with the mental situation of transness.
You are looking at this from the perspective that wants to categorize based on your distinct values. You want to determine effectively whether a trans person is effectively really "entitled" to being called what they want to be called. The tagline "a trans woman is a woman" is unfortunate because it is a slogan that doesn't give the full account of why it is important and the whole situation is muddied by the fact that the wider concept of gender performativty actually has almost nothing to do with what trans people are actually experiencing.
Logical fallacy wise stating that something was determined by historical precedent is also a fallacy. It's called an "appeal to tradition".
What is happening culturally with trans people is an attempt based off the findings of years of intensive psychological research to create sociological tools to ease the burdens of a minority population. It might be effective to conceptualize this as language being a technology and that technology effectively being applied as medicine. The people who value the comfort, and quite frankly an expanded lifespan, of trans people adopt this framework but, because to be successful it requires participation. Ideally they teach other people the reasons why it's important to the point they will happily adopt it but that isn't wholly nessisary. As long as someone is treating say, a trans man by using his name and pronouns and not assuming his behaviour to conform to feminine restrictions then effectively the "medicine" works. Hence "trans men are men" ie treat a trans man - as you would a man. An expectation squished into a narrow confine with all nuance removed.
The reason "biological woman" expressly doesn't work is what trans people are responding to is almost completely their own biology. The cultural stuff about gender is kind of just layered on top. What they are responding to when someone uses pronouns is their own physical state. Say you kept calling a trans woman "he/him" what that is doing isn't impacting some attempt at manifesting some spiritual form of womanhood - you are demonstrating you are veiwing her body, seeing phenotypic masculine characteristics and reporting them back to her. Her brain is wired to pair that with a stress reaction. To her those parts of her body are things she desperately wishes doesn't exist because veiwing them, interfacing them sometimes touching them - is abhorrent. What you are doing when you use people's pronouns is effectivly creating a mirror of words. The only question is whether that mirror of words is kind to the viewer. Does it reflect the things that soothe or does it reflect the things that cause strain? That's something the speaker of those words controls because the trans person is powerless in this regard which mirror the speaker will offer them.
Saying "biological woman" aloud in front of a trans woman is effectively indistinguishable from the mental reaction you would create by calling her a man. You are reminding her that both to you and probably to herself that her body is a compromise she has to live with. She's effectively doing everything she can but it will never be enough not just for you... But for her.
When the compromise of living in an imperfect situation becomes too burdensome not living becomes a more viable solution. It won't kill every trans person on it's own but paired with other factors it tips the scale an outsized amount. The reason the historical definition of man and woman is the way it is is because as a population trans people were veiwed as deviant, weird, lead by devils into perversion and a public nuisance and them being miserable was culturally a perfectly fine outcome. Them being miserable in private until they were overwhelmed and killed themselves or being treated as circus freaks- not really a problem.
In modern day we generally hope for better.