If I'm not mistaken, such problems would contain some enumerated hypotheses - point peaks in a well-defined parameter space - and the NOTA hypothesis would be a uniformly thin layer over the rest of that space

Yep.

Can't tell what key features the data-generating process must have, though.

I think the key features that make the NOTA hypothesis feasible are (i) all possible hypotheses generate signals of a known form (but with free parameters), and (ii) although the space of all possible hypotheses is too large to enumerate, we have a partial library of "interesting" hypotheses of particularly high prior probability for which the generated signals are known even more specifically than in the general case.