I agree with you on the uselessness of definitions that don't distinguish anything, and I think your conceptualisation of a social problem as being something that requires multiple people to interact (like a scam) is useful, admits a very clean boundary, and deserves a name. I would call such a problem a purely social problem. But I think there is a broader concept of "social problem" that makes sense too, even though it lacks such sharp edges.

I think it's reasonable to say that a problem is a social problem to the degree that its severity as experienced by one person depends on other people's behaviour.

If I accidentally drop a rock on my own foot, this seems to be obviously not a social problem. But if I am more likely to be carrying a rock in the first place, or less likely to be wearing protective shoes, because of how society is organised, then to that extent I claim that it is a social problem. This is not an abstract example: Over time, changes in society's attitudes to dangerous kinds of work have directly, and indirectly through health and safety legislation, led to massive reductions in workplace harm since the Industrial Revolution.

Under this rubric, all it takes for a problem to be social is for it to be possible to imagine that society being organised differently would lead to a different level of suffering. Does this lead to nearly all problems being classified as social problems? Yes -- but that is not a problem in itself, that is just an accurate picture of reality! It is still useful -- indeed much more useful -- to place problems on a spectrum of social-ness; nothing "collapses" unless we are determined to make a black-and-white distinction.