Firstly, "by programmers for programmers" is meaningless. Anyone who can write code is a programmer -- maybe an amateur programmer, but a programmer nonetheless. By this definition everyone on SO with posts under their name (except spammers) is a programmer.
Would it be possible to add a filter that only shows questions asked by users who have proven their ability to actually provide answers?
No. Why? You are effectively deprived of a good source of community members. And you're destroying SOs worth. Relatively few people start on SO by answering. Most ask questions first. Some of them go on to become great answerers. Some don't.
Either way, a large portion of your community is from people who asked first. I started off on Physics.SE in much the same way: I asked a question, liked the place, and stayed over. However, the only reason I stayed was that I quickly got comments and upvotes helping me. If I had gotten crickets, I would have just left.
Besides, a lot of the questions seem to come from these people whose questions get ignored.. If I'm stuck and need help, I'm not going to spend time answering stuff till I reach the threshold. I'll pack up and go elsewhere. I'm pretty certain that almost everyone will use this filter if it was implemented, which means that SO's value in "getting good answers" will be diminished.
In the end, we would end up diminishing a massive source of community members and a source of questions. That's not good.
Stack Overflow is not, and should never be, an "elite club". Elitism in online communities always causes big issues and harms the community. You see a bad question? Close it and comment. Don't ignore it.