there is the 3rd option that they want the people to go to the restroom that conforms with the genitalia that the person was born with, but your straw man rendition of the argument only works if you ignore the obvious.
it's a tough, novel issue for EVERYONE to deal with and anyone on either side that thinks they have the 100% absolute correct answer is too shortsighted and morally blind to consider. i think a 3rd bathroom would be ideal but a legislative nightmare. consider that when new ADA regulations go into effect, businesses that serve the public have 180 days to comply. there is no granfathering in to get around ADA. hopefully, we all see the sense in protecting the handicapped, even at the inconvenience of the business owner, even if we don't agree with a specific regulation and even if we think that the regs should only apply to new construction or when a property is remodeled. having a third bathroom isn't as easy as lifting a sink, widening a door or adding handicapped bars. so while it is the ideal solution, it's not practical. on the flip side (e.g., letting transgenders use the bathroom of the gender they identify with), do you want adults with p**ises changing in the same locker rooms as girls? should the former's gender identification matter when the question is so bluntly stated? already people are going on youtube to prove the point this way.
so... while we should be sensitive to a person's psychological well-being when afflicted with gender identity issues, we shouldn't ignore the very real concern of the rest of society. it's a balancing act. i for one am happy i'm not a politician having to decide what to do...