It heaps on a lot of theoretical musing without providing either empirical evidence of practical relevance or providing a theoretical guarantee along the lines of "as long as the model is within delta of [something intuitively verifiable] it will have [useful property] with p>1-epsilon."

For what you're trying to do, there is a relatively solid baseline, namely doing something trivial to first-order logic (e.g. assigning a dimension to each Herbrand formula or ground term or whatever) and e.g. turn generalized quantifiers into tensors (with some mathematical plumbing required). This would allow you to reuse model-theoretic semantics with a serving of tensors and vector spaces on top, and it would do at least those things that generalized quantifiers can do. You could then argue empirically for some kind of finite-dimensional approximation to that, or expose neat theoretical properties from a formal viewpoint.

As is, you don't do any of that. It's more or less like going into a hardware store, getting some pipe and toilet bowls and making an abstract sculpture without ever talking about the house for which you want to provide the plumbing.