So the way I see it, "living as a girl"* consists of two things: Acting as one thinks girls should act, and being treated as one thinks girls should be treated. But gender stereotypes are increasingly deprecated. There are very few places in society where we can find a clear, binary demarcation between men and women. Very few places where law and custom still firmly uphold the traditional view of sex and gender.
So while there's more to it than just "access to women's spaces", I think that is the predominant factor, by a very large margin.
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The whole question gets really weird, really fast, though. Does living like a girl mean wearing dresses and putting on makeup? But men can wear dresses and put on makeup, too. Nothing wrong with that.
Actually, a lot of communities do frown on men wearing dresses and putting on makeup. This is good for transwomen, because it means there's still a gender-stereotype line that they can cross. On this side, they're living as a man. On that side, they're living as a woman.
But they're not living entirely as a woman. If they're mannish enough, their community is still going to frown on them for cross-dressing. So even though they're acting like a woman, they're not being treated like a woman. Their community needs to treat them like a woman, not like a cross-dressing man. Really, their community needs to stop frowning on male-looking people who wear dresses and put on makeup.
But this then erases the clear gender-stereotype line that the transwoman crosses to achieve their goal of living like a woman. If there's nothing wrong with men wearing dresses and putting on makeup, then there's nothing to be gained in terms of lived gender identity by doing so.
Eliminate all these gender stereotypes, and what's left? Women's locker rooms. Women's sports.