Actually, I don't have a problem with it. Do you know how hard it is to have an older autistic child who cannot go into the bathroom with you? And yet, he's independent to the point where taking him to one of those single-toilet-and-sink "family" restrooms is too much of an invasion of his privacy (and mine, if I need to go).
What I do have a problem with is stalker, rapist-y men hiding out in the women's bathroom, and then pretending they're suddenly having an identity crisis if and when they're confronted.
In my opinion, the whole "let transgender people use the restroom of the gender they identify with" argument isn't even about transgender people. Because guess what. Truly transgender people make a pretty big effort to pass and you're probably not going to know. It's not a lifestyle I agree with, but I'm just saying.
I think it's more about expectations of privacy and decorum. Our culture may be changing, but many of us are uncomfortable enough trying to poop when there is a one-inch gap between the partitions in the stalls, and about a foot and a half near everyone's feet. It's hard enough to try to get "stuff done" when toddlers peeping under the stall can see you. We'll all hold it in and die if actual regular guys could see us dispose of our used sanitary products (thanks for the image? you're welcome).
So. If we're changing things, how can we assure people they have a reasonable expectation of privacy and safety? Because 1. I don't think transgender people are going away, and 2. I don't think it's unreasonable that we feel safe and have at least a reasonable amount of privacy in the bathroom.