Hi!
I used to fly with my dog a lot, on all different carriers. Never had a problem.
cliff
NBO
PS-Pilots are trained to know about carrying live animals, and they are notified if live animals are carried, so that they can more safely transport them. An emergency may apply where your pet would die, but that applies to you and the pilots also. If you are worried, drive a car, and then you will be less safe, but you might feel better, because you are in control of the car.