It is from the word "dépanner", not from the word "panne" directly. "Dépanner" evolved from "panne" to mean just in general, helping someone out temporarily.

A tool that would dépanne you would be a tool that would do the job poorly but well enough for now. This is how the name is meant to be understood, a small store where you can buy like eggs, beer, milk, bread (it's a convenience store), maybe batteries, but not a full grocery store or pharmacy or tool place.

Yes, it "dépannes" nearby, instead of making a trip to the grocery store.

Maybe it arose from originally being kind of fix it stores that would repair stuff, I don't know.