Of course they care. (law suits and bad publicity) The problem for them is not the plane ride as far as I know. I believe the temperature is controlled on the plane. It is the handling of the pet before the plane leaves and after it lands. With Continental then United the pet would come out to the plane in an air conditioned truck and then also they loaded the pet last just before the plane leaves. When the plane lands the pet is the first off and placed in a air conditioned area. Other airline to the best of my knowledge do not do this and that is why they will not fly a pet above or below certain temps. They have a date range that they block out because of the likelihood of high temps.
With Delta I've seen my pet out on the tarmac for a half hour or more a few times. That of course would not do during warm months. United does not do this. Of course you pay for this service. Also United has a kennel in Houston with people trained to take care of pets and board them overnight if necessary. Now I stopped using United because they made several mistakes which caused me a lot of inconvenience but never endangered my pet. One time they sent me to Cargo on a day it was closed. Then they compounded the error by telling me just to bring the pet the next morning and that they had made arrangements for him to fly as military baggage. I told them that would not work but they insisted so I got the person's name and number and sure enough the next morning when the supervisor for the ticket counter said he couldn't process the dog I called the person and handed the phone to the supervisor. They talked then then booked me on a flight later that day. I had to stay at the airport all day, then take a cab over to the cargo section check in my dog and return back for the flight. In Houston I had to stay overnight and my dog was put in their kennel. the return trip several months later wasn't much better. All of that upset me a little you might say and that ended my flying my pet with United. Still if I had to I would use them again but only if I had to.
One time I researched going through Mexico with my dog and if I remember right there is a quarantine and other problems which immediately ruled that out. Maybe that has changed but I doubt it. Mexico is much worse to deal with than Belize which is really pretty good with pets. In the circumstances described I would contact United Pet Services, I think they called it "Pet Safe."