As someone who has had many of these encounters, I've never had the police help. At best they'll show up a half hour later and take a report. The police were never "there" in any meaningful sense.
> because (for all their flaws) they are still less trigger-happy than a frightened gun-owner who feels threatened by an encounter with some strange-acting person
Time after time, this has proven to not be true. When a frightened gun-owner reaches for his weapon, he knows he is risking all his money and probably his freedom if he's found to have fired it when it wasn't necessary. And so they tend to use some discretion. Cops do not risk the same, and so they shoot people all the time when their life isn't threatened.