It was never a problem before a vocal minority declared it a problem of some sort. There never was a 'genital check', and no one proposed such a thing. Why be silly? Your birth certificate does indeed indicate your physical sex.. so call that an 'ID card', if you like. If your actions offend someone (you appear male to the reasonable person, but enter the female rest room - or vice versa) then it is you that needs to change behavior, not the rest of the world. Go into the bathroom that you are accepted in by the community that is sharing the facility together.
No. Community sentiments don't identify people. Unless there is compelling reason, insanity or extreme physical capability requiring special care, a person is allowed to identify as she feels comfortable. While communities may have men and women bathrooms it doesn't determine what one believes is their sexual identity. At some point one has to go on trust in communities of people.
Call me the next time a transgender person attacks your daughter in the little girls room. It will be a first.