I have a friend that trains service dogs (she worked with a group in TX that trains mobility and hearing dogs) and she HATES stuff like this.
According to her, ESAs don't have to go through rigorous training or temperament testing to prove that they will be safe in public and most are pets without special training. The designation of ESA refers to the person/patient needing the animal for emotional reasons, not the qualities of the animal itself. Legally, they are not required to be allowed in public places.. I believe the only legal requirement is for landlords/housing, and airline travel and places of employment (with restrictions on safety and the other staff taken into account).
As far as I know, small dogs and cats are allowed in cabin if in approved carriers that fit under the owner's seat and you have to pay an additional fee to allow them to travel that way. If the animal and carrier don't fit under the seat, they have to fly in cargo in an approved carrier and this prevents larger dogs from being in cabin just due to size.
IMO a good way for airlines to crack down on this is to require any ESA owner having an ESA (even a small one) outside of a carrier to carry and show proof of liability insurance on the animal in the case of a bite. Liability for animal bites is expensive and I'm sure if someone legitly needed an ESA, they'd pony up the cash for it and if not...that would negate the point of scamming the system to get your animal a free plane ride (it cost around $150 to have your animal fly in cabin I think and insurance would be a lot more than that annually). I'd be all for proof of insurance being required for service dogs as well (that enter public places where dogs aren't allowed). This would cut down on scammers and I'm sure animals trained with legitimate service dogs places would be able to get insurance at a much lower rate. Of course, this could just open up another can of worms in terms of scammers (bogus insurance companies, etc.) but there's a way around everything.