I'm far from a UA apologist, and I have a child with medical issues myself, so I'm sensitive to issues around travel with special needs kids.
That said, from what I'm reading from the account, the mom was in the wrong here, except maybe as to the FA's attitude. She should have checked with UA and gotten it cleared.
One element that i didn't see addressed is that an unrestrained 3 year old who is unable to support themselves is a danger to other passengers in the event of an emergency or turbulence. No way mom could hold her in the event of an emergency landing.
UA is guilty of a lot, but I'm not feeling this one.
Greg