Wow, I would
never send my 5-1/2yr old into a public mens restroom alone. Sorry, there are too many weirdo's and whacko's out there, and I am
not going to be lulled into some false sense of security that they are not at WDW too. Actually there are probably more there than people even realize.
And I also fail to see what is "disgusting" about it. After all, women use stalls so there is at least a little bit of privacy there. When I have to take ds into a ladies room, we do our business as quickly as possible, wash hands, and leave. There is no horse play allowed. And as long as we keep it that way, we generally have no problems and no one bothers us about it.
When at WDW, we see lots of little and some big kids of the opposite gender in the restroom. I've personally never thought anything of it, as long as they are behaving themselves.
edit to add: there are no "family" restrooms at WDW. The ones people are usually thinking of are "companion" restrooms. They are actually Handicapped accessible bathrooms for people that either can't fit their eqmt in a standard H/C stall in the regular restroom, or need a companion (sometimes of the opposite gender) to help them out, so it wouldn't be appropriate in that case for a man to go into a ladies room or visa-versa. Anyone
can use them but that is really what they are intended for.
Another option may be the bathrooms in the baby care centers. They have regular restrooms there. Since this facility is staffed, and the bathrooms are very small with only one entrance and exit, I would feel more comfortable sending my ds into one of these bathrooms alone. But there is only one baby care center per park, so that is not always convenient.