I don't think that the examples that you have with the waitress and the mechanic are good comparisons, because they are "cute" after solving your problem. More accurate examples would be the waitress or mechanic saying "Oops, sowwy! Something went wrong." and leaving, without solving your problem.
I'm also on the side of being professional while the problem still stands, because the user is most probably annoyed at the situation, and wants you to take it seriously. You can be cute after the issue has been resolved.
The other article you mention comes from the same website: https://rxdb.info/offline-first.html. It's in an opinion section, giving the arguments from both sides.