If you use a model where the dependent variable is a function of a single variable then you are doing standard nonparametric regression and you can use kernel regression, splines, local polynomials, etc. The problem is often that you don't know what variable you want to regress on or more than one variable is important.
You are right that the basic motivation for projection pursuit regression is the curse of dimensionality. Unfortunately, I think your question stems from a flawed premise. That is, there is no equivalence between finding structure in a projection onto a subspace and finding structure per se. Just because your data has structure does not mean projection pursuit regression will find it. If there was such a guarantee, then there would be no curse of dimensionality. The method may still provide good results for your problem though. The basic question of course is whether or not a relatively small number of directions captures most of whatever is interesting in the data. The nice thing about projection pursuit regression, compared to say, principle component regression, is that you can learn nonlinear relationships in these directions. In practice, it is somewhat rare that a practitioner would actually have an a priori reason to think projection pursuit regression will learn the true model. The directions and associated smooth functions are normally difficult to interpret and the results are accepted if they perform acceptably in out-of-sample prediction. It is hard to do much better for general high-dimensional problems.
I have a (non-comprehensive) list of suggestions for you.
1. Try doing PCA on the data first. If any of these directions seem sensible then you might be better off doing one-dimensional regressions on these.
2. Try generalized additive models, which do not project the data. If you do not think that linear combinations of the data are interesting then avoid regressing on them.
3. You may be able to throw away a lot of your variables without losing much. If you can pare down the number of variables then projection pursuit regression or generalized additive models may give better results. There are quite a few methods for doing this, e.g. sparse additive models, forward/backward stepwise selection, and using bandwidth selection techniques to "smooth out" uninteresting variables.
I hope this helps.