"Why wouldn't they process?"

Solution 1:

The modal verbs will and would have two senses, like all modals:

1. the deontic (social -- permission, obligation) sense of 'be willing to', as well as
2. the epistemic (logical -- possibility, probability) sense of 'be predicted to', which Latinate pedants call "The Future Tense".

Sense 1 is what appears in negative uses of would, and also in the hypothetical uses of both will and would -- i.e, this sense is most common in Negative Polarity contexts, like the NPIs in this puzzle.

• I won't do it! = I am unwilling to do it!
• They would not listen to him. = They were not willing to listen to him.
• If you would see it my way, = If you were willing to see it my way,
• If she will accept his apology, = If she is willing to accept his apology,

Solution 2:

Yes, it is a "well-established" English expression.

The Modal Verb "would" is sometimes used to talk about something that somebody refuses to do.

Ex. He just WOULDN'T listen.

This is why it is also used to complain about situations and about things that you've repeatedly tried.