I'm a little unclear about what EY's intended philosophical purposes are in this context

As I read him, he mainly wants to make the point that "simplicity" is not the same as "intuitiveness", and the former trumps the latter. It may seem more "humanly natural" for there to be some magical process causing wavefunction collapse than for there to be a proliferation of "worlds", but because the latter doesn't require any additions to the equations, it is strictly simpler and thus favored by Occam's Razor.

I think you might be conflating configuration space with the Hilbert space of wavefunctions on configuration space.

Yes, sorry. What I actually meant by "configuration space" was "the Hilbert space that wavefunctions are elements of". That space, whatever you call it ("state space"?), is the one that matters in the context of "wavefunction realism".

(This explains an otherwise puzzling passage in the article you linked, which contrasts the "configuration space" and "Hilbert space" formalisms; but on the other hand, it reduces my credence that EY knows what he's talking about in the QM sequence, since he doesn't seem to talk about the space-that-wavefunctions-are-elements-of much at all.)

If this is your standard for lack of rigor, then perhaps QFT hasn't been rigorously formulated yet, but the same would hold of pretty much any physical theory

This is contrary to my understanding. I was under the impression that classical mechanics, general relativity, and NRQM had all by now been given rigorous mathematical formulations (in terms of symplectic geometry, Lorentzian geometry, and the theory of operators on Hilbert space respectively).

Maybe QFT isn't rigorously formulated according to the mathematician's standards of rigor, but it meets the physicist's lower standards of rigor. There's a reason most physicists working on QFT are uninterested in things like Algebraic Quantum Field Theory.

The mathematician's standards are what interests me, and are what I mean by "rigor". I don't consider it a virtue on the part of physicists that they are unaware of or uninterested in the mathematical foundations of physics, even if they are able to get away with being so uninterested. There is a reason mathematicians have the standards of rigor they do. (And it should of course be said that some physicists are interested in rigorous mathematics.)