By this framing, everything is a construct. And fine, there's a long, fruitless, metaphysical argument we could have about that. But it's an asinine argument when it comes to communicating in any useful way.
We can argue all day about whether a chair is, in fact, a table, or we can admit that we know what we mean when we say "chair" and "table" and not waste each other's time pretending that the extremely rare cases where it's not perfectly clear which of the two something is prevents us from ever knowing.
No. I don't think woman is more than biology. "Woman" is a simple, well understood word that should be exactly as uncontroversial as lioness or hen. Mother is the term for a female parent. Again, uncontroversial until a few people decided to make it so.
What you're talking about there is feminine stereotypes. And if you're arguing that conformity to feminine stereotypes does or does not make you a woman, there's a whole other conversation we need to have.
You make a fair point about adoptive mother vs mother. But my point isn't that we should constantly say "trans women." It's that, just as adoptive mothers are different in some meaningful ways to biological mothers, trans women are different in some meaningful ways to women. And it's not hateful to admit that on the occasions that it's relevant.
If adoptive mothers wanted to have their children treated as if they were genetically related, ignoring any risk factors the child had from their birth parents, that would be deeply irresponsible. We can call adoptive mothers, mothers. We can call trans women women. As long as we don't lose sight of the differences between the two.
Sometimes feelings have to give way to objective reality.