In this scenario of yours where would this doctor have their "airplane" at home? In a airfield?
For an emergency I would suggest a professional-driven ambulance with flashing light, or an helicopter if it’s remote if your wallet or country medicare afford it.
Fire your transit leadership. Trains should be running every 5 minutes all day every day (and also have a backup option for night service while you do track maintenance). Incompetent leadership isn't doing that.
For a bus you are sometimes forced to do cost/benefit and run as little as every half hour, but if you have trains that means you have enough demand to run every 5 minutes (if you don't that means you shouldn't have a train at all - a bus is cheaper).
It isn't just doctors that sometimes have an emergency. People have places to be, trying to figure out when the transit system is coming shouldn't be a problem.
We're in a discussion about flights replacing 100/400 miles trips.
If fear of people were the #1 reason not to go on public transport where you live, I'd be deeply worried for your culture
My SO takes the train everyday to go to work, literally every day (we live in Colonna, near Rome, and she works in Rome center) and this has never happened to her.
There are downsides, trains might be full at peak times like 7.30 AM and you get to stand in middle of many other people, but what would the alternative be? Twice the amount of time and a magnitude of order more of stress to get there by car? She doesn't even need to buy a car, we only own one in our household and it sits in the garage most of the time.
On the other side it’s easy to picture why a regular driver would be stressed in a train : you’re not in control.