I definitely agree with you on these two points (involving "the processing of...is quick" and "...easily and quickly processed"). Both of these highlight the association between "quick" and the action of "processing", which is what seems to be missing in my original sentence. What still puzzles me is why "the processing of...is quick" seems much better. "Seeming" is such an unsatisfactory justification when it comes to grammar!
I think this is due to the fact that, being a verb, the V-ing form will not accept certain adjectives, but a noun will, and "the processing" is a noun. Compare: The assessment was quick vs. Assessing was quick, The movement was quick vs. Moving was quick, The response was quick vs. Responding was quick, The answer was quick vs. Answering was quick, The solution was quick vs. Solving was quick.
I quite like this test, though I would actually disagree with this last point about its showing the adjective is related to both the subject and the following infinitive. If anything, I think it serves to verify the link between the adjective and the subject and not between the adjective and the following verb (which is why "quick" fails the test). In other words, in all of your black examples, the test shows the suitable association between the adjective and the subject. However, with "quick", the association is shown to be invalid. What do you think?
Could we try using your test with the following example (based on a sentence I just found on a WHO webpage)?
– "Children are urgent to vaccinate." (Original sentence: "It is less urgent to vaccinate children [than vulnerable age groups].")
I think this too is ungrammatical, since it is the vaccination that is urgent and not the children as people/beings.
As I said at the beginning, my understanding is that the adjectives in question are related to both the subject and the infinitive—it's not that the sentences are just easy, but easy to process, or that somebody is pleasant, but pleasant to speak to or to work with.
In:
- Children are urgent to vaccinate
"urgent" does not combine with the infinitive. Suppose we say:
*Children are urgent patients to vaccinate
where "patients" is the generic word of the test I proposed. Clearly, this does not work. But this does:
- We have / There are some urgent patients to vaccinate
because there is no relationship between the urgent nature of the patients and the fact that they are to be vaccinated.