If you can pay something that's related to the niche, it shows you have expertise in the niche.
For example, if you can pay $100/month for a business forum, than that shows you are at least somewhat serious and interested in business, and more likely, you are already successful at it.
If you can pay via open source contributions to the OS community, and that's a requirement for a software community, it shows at least you can code decently.
It doesn't have to be a "Do X" type of test either, it can just be an interview to hear their experience, their attitude, etc..
>Or, how does inability/unwillingness to pay or prove yourself in some other way relate to an inability to make worthwhile contributions?
You design a test to filter out people that you don't want in your community. If people fail that test, then it should already signal an inability to make worthwhile contributions. It might give false negatives for users that would have been good, but the test should not give too many false negatives.
No it's not. It's a problem all large and free communities face.
Try looking for private communities, and the quality is much higher.
To get into these types of communities, you have to pass an interview or maybe pay monthly (I don't belong to these types, but I've heard good stuff about their quality).
e.g. want a good business community?
Maybe have a fee for $100/month, so you have people who can afford that join.
want a good programming community? Maybe have open source contribution requirements.
Or just interviews in general will filter out a lot of crappy users.