The important distinction people seem to miss is that emotional support animals are protected by the Fair Housing Act (FHA) - NOT the Americans with Disabilities Act (ADA).
If you have a legitimate need for an ESA and a letter from an actual doctor you do get some useful rights in your home. A big one is that you don’t have to pay pet rent/deposits (as much as $600/year in some apartments).
The fair housing act rights for your emotional support animal end when you take it outside your home. They are not service animals. I agree it is incredibly lame when people don’t know the difference and bring their pets places they don’t belong.