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✲He stopped finding the key. [achievements] (1)
He stopped painting the house. [accomplishment] (2)
(The Cambridge Grammar of the English Language)
It seems like reason (1) is wrong. is finding is an event occurring at a point in time, while painting is a durative process. Isn't there any duration at all in the word find? If there isn't, how do we have to correct the first sentence for the meaning of stopping the activity of finding the key?
Your analysis is correct: find in the sense used here is not durative*, so it cannot be terminated with stop.
In English we do not use find as an activity verb; instead we use look for and seek. In fact, find is itself a terminator for those verbs—as the saying goes, “You always find what you’re looking for in the last place you look”.
Note that look for and seek are what your grammar probably calls atelic verbs; they denote the activity of trying to find, but do not include the successful achievement of finding.
If you want to signify ending the activity without success, you may say “He stopped looking for the key.”
* In another sense, however, it may be durative, as when we say “I find that difficult to believe”.
I agree with you. If you refer to only one object, you are searching for, then either you find it or not. There is no process of "finding" one object, at least it seems unnatural, because you would have to lose it again in order to find it twice.
If we talk about many objects, then I believe, there should be a use of "finding"? But I don't know exactly. For example, in a game of searching keys, as a non-native speaker I would perhaps use "The process of finding the keys".
