The letter I regret is worthless. The airline cannot protect you from litigation and in the case of treating a US citizen you could have a decade of dozens of court appearances and even the threat of being kept in the US
However most of us don't expect immunity and we carry our own professional indemnity.
Over 30 years I lose count of how often I have paid for a seat only to spend hours sitting on the floor looking after some passenger, missing food and films, sometimes to be required to accompany the patient to the hospital. It was normal practice to receive a letter from the airline offering a refund of my ruined flight or a couple of free business class seats in compensation
Slowly slowly airlines realised they didn't have to and now the norm is not even a thank you. Many doctors nowadays refuse to volunteer and it is hard to blame them. That is how much airlines care about their passengers. True, a number have contracts with telemetry medical services and some are very good, but they will play safe and I muse over how much the resultant diversions and returns cost compared with a free ticket in the old days