As ESAs are a US-specific thing, I assume you're located there and are looking for advice for this region. Apart from US and Canadian airlines, ESAs are not accepted on board anywhere else in the world (except on direct flights to and from the USA where airlines are legally forced to accept them).
As previous posters have mentioned, rules are being tightened across the board, and an "ESA letter" won't cut it anymore. This is the case for all the network airlines -
American,
Delta and
United all have become more restrictive and increased their requirements considerably over the last year. See the announcements of restrictive changes that AA published in
2018 and again in
2019.
Check out the policies linked above and then decide which airline will suit your needs best. If the networks of
Southwest and
Alaska will work for you, add them, too. Stay away from the ULCCs like
Frontier if your dog is more than pint-sized as seating is far too tight for larger animals to fit comfortably - the only exception to this is
Spirit with its
Big Front Seat.
Personally, I can't report bad things about AA - I`ve been flying them for a decade and hundreds of thousands of miles with a service dog in all classes of service and recent changes didn't impact our travel experience much at all. Crews and ground staff in general have been treating us very well, and they remain our airline of choice.
Especially if your dog is larger, network, aircraft and seat selection is important. More important than minor acceptance policy differences. What you need to be looking for is floor space, more floor space and even more floor space (without encroaching on your fellow passengers) followed by fewest connections. Familiarize yourself with the routings, aircraft and seating configurations airlines are using on the routes you intend to fly with your dog, and select your preferred airline according to what will work best. Example: If you want to fly between Miami and Los Angeles, AA flies this route nonstop (always preferable with animals) and one of the daily departures is on an internationally configured four-class 777 widebody with a PE cabin - the bulkhead seats in which are the best place to be for a large dog apart from the eight-seat F cabin.