You have a great specific story and a nice takeaway. I think there are a few issues that worked against it on Medium.

First the title. It tells me this is a story about a time you were choking. The subtitle backs that up. This might grab readers who have also experienced choking but most readers on Medium pass right by personal stories until they already know and love the writer or it is an experience they have had or resonate with for some reason.

For me the real strength of the story lies in the juxtaposition of concern for your parents comfort and safety when you are the only one who faced death. This is more universal. I would work that into the title.

The other really big issue is the opening. To keep readers on Medium you have to come on strong. Hit your theme hard right up front or build suspense by starting with the middle.

I would consider taking the paragraph that begins, "I stared into the distance at the pretty summer-time snow ski runs and actually thought to myself; this is IT. " and move it all up to the very beginning. You can then go back to the reason for your trip etc.