Megan
The question you're avoiding is:
Would the dog have died if it hadn't been placed in a minimally ventilation overhead?
Neither you nor I can answer that question. The fact that certain airlines ban their carriage and vets refuse to sign paperwork to allow them to fly should be enough information as to the risks posed. The woman had gone through the necessary process to carry an emotional support animal? If not, I don't see that the condition of carriage allow the animal to be in the cabin.