It doesn't change the sex of a person, it matches their body to their mind, so to say.
Best way I can put it is like this.... A fully transitioned female in a female locker room shower would go pretty much unnoticed... where as a transgender female with male parts in a female locker room shower would not only be greatly noticed, it would more than likely offend many to say the least.
I see what you mean. I agree that a sex reassignment surgery is not what would change the sex of the person transitioning, if one would want to argue their sex is changed at all. I will always consider the hormone therapy to be the biggest change, because hormone therapy changes so much about a person. Including their muscle and fat composition, their appetite, their libido, their responsiveness to drugs/medicine, etc.
For example, did you know that many trans women on estrogen cannot become erect, and those who can usually not enough to penetrate a partner? While they still experience pleasure from stimulation, they can't get hard or penetrate anymore. The opposite is true for trans men on testosterone. The clit grows (usually between 1-2 inches, but sometimes 3 inches) and begins to act as the penis would, with the person becoming erect upon arousal, morning wood, etc. Some can use their enlarged clit to penetrate partners.
Hormones are what dictate what the body part does and acts like. A sex reassignment surgery just changes what the body part looks like.
Also, I find the notion that the sex reassignment surgery changes the sex silly, because, imagine if a biologically and mentally female woman chose to get phalloplasty (construction of a penis) without breast removal or testosterone. Phalloplasty doesn't involve the removal of the ovaries/uterus, or even the vagina (that's a separate surgery that's usually done at the same time if the patient wants it, but many choose to keep it), so that theoretical person would certainly still be female.
On the contrary, a trans man with with top surgery, hormones, etc. would stand out in a woman's bathroom just as badly, if not worse, than a transgender woman without a sex reassignment surgery. Hairy chest and legs, a beard, and (for all intents and purposes) what appears to be a micropenis. I don't think most women would appreciate seeing that in their locker room.
This is why I see steps in transition as being arbitrary and pointless to obsess over. So many people focus on transgender women, and completely gloss over how powerful testosterone is in trans men. People also gloss over the fact that secondary sex characteristics are so obvious.
So what if the trans man (on testosterone without sex reassignment surgery) has no penis? He still has the body of a man in every other aspect. And the strength of one, too. In some cases he could even penetrate a woman. So clearly the "people belong in their biological sexs' bathroom unless they've had 'the surgery'" assertion is a bad one.
Whether someone has had bottom surgery or not says very little about that person.