Find items that are in at least $k$ of $n$ sets
OK, I think I found something relevant: this paper mentions a "T-occurrence problem" in section III (p. 2) which is exactly our problem (where $T$ is what we called $k$), hidden behind some domain-specific jargon. The ScanCount algorithm that they propose is the naive approach I suggested in my question. The MergeOpt algorithm is a generalization of the binary search trick. Their main proposal (DivideSkip) is a combination of this binary search trick and a different trick (MergeSkip) to skip multiple values. It even seems that experimentally the clever approaches are much better than the naive approaches (look at the "No filters" column in page 8, the filters are heuristics for their domain-specific stuff).

This can be combined with David Eppstein's trick to make multiple binary searches in $S_2$ more efficient, and with the idea of using interpolation search instead of binary search (an idea from this other paper from the same field).