quote from The New Quantum Universe by Hey, Walters:

is that an accurate statement?

does it imply that there's a complete non-link between even the simplest intuitive qualitative descriptions of quantum theory and the mathematics of it?

The statement by Hey and Walters is misleading. The Feynmann diagrams are a condensed way of setting up the math. Each element in the diagram (input line, output line, exchange line, vertex) corresponds to a factor in the propagator integrand, so you can set up the math directly from the diagram.

The math will work either as the description of an antiparticle traveling forward in time or as the corresponding particle traveling backwards in time. (think positron, electron). QED doesn't say which is the case. Feynman apparently found it easy to think of particles moving backward in time, but most physicsts prefer to think in terms of antiparticles.