Are there any ‘high-class photographers’ that can produce good results for every single shot, so that the shots don’t require post processing?
If not, then is it possible to become a top photographer without knowing how to edit?
It is at least partly true of some of the busiest photographers. It has nothing to do with the photographs not being edited, and simply means the photographer isn't the one doing the editing. Just for example, during the Olympics, the guys doing shooting just shoot. They have at least a couple of cameras, and an assistant who's responsible (among other things) for ensuring that that they always have a camera with memory space available.
When one's full, they trade cameras with the assistant. The assistant usually won't handle the editing either -- his job is primarily swapping cards and (if necessary) lenses. The cards he just gets (via whatever means) to other people who download the pictures, bring the cards back to the assistant, etc. Somebody eventually edits the shots, but there are probably going to be three or four layers of other people in between the photographer and the person who does the editing (and, just to keep things interesting, a fair number of those people probably have something like "editor" in their job title).
That, of course, is a fairly extreme case, but the same general idea remains in a lot of other cases as well. The photographer handles shooting, and leaves the editing to people who specialize in editing.
If what you intended to imply was shots so good they didn't need editing at all, then it's mostly nonsense. To even stand a chance of being true at all, you have to start by defining "editing" pretty narrowly. Just converting a raw file to a "picture" at all requires a fair amount of "editing". Except for the very most simplistic conversion (that nobody would willingly use) there are quite a few "editing" types of input into the process of converting a raw file into a picture (e.g., sharpening, adjusting brightness, contrast, white balance, etc.) In this case, there is no really "unedited" version of things -- at most, there's the "default" version, which basically just means leaving the "editing" to the people who wrote the software (and since they obviously can't see the picture in question, their decision(s) are rarely optimal for any given photograph).
If you mean there are no localized edits, and all that's done is "global" editing (the aforementioned adjustments to overall brightness, contrast, WB, etc.) then it starts to have at least some possibility of being true at least part of the time -- but only if you make it a tautology, by defining "high class photographers" as photographers who don't do any of that sort of editing. Otherwise, it's still pretty much nonsense.
Edit: in response to edited question: By you added definition of "high class photographers", I'd say there simply is no such thing. I've known a fair number of excellent professional photographers. I doubt that any of them would consider publishing or displaying even 10% of the pictures they took. For most, the number is probably closer to 1%. Part of that is from experimenting and such, taking pictures they don't expect to be worth much. Part, however, is simply having high enough standards that even slight imperfections are enough that a picture won't be used (at least as-is). The reality, however, is that if even half your pictures come out "good", you either have very low standards, or you're not challenging yourself much (if at all).
No, that is incorrect.
However, some photographers do avoid digital post-processing, and some still shoot film.
To say that all "high class" photographers would avoid editing software is far to general of a statement though.
A "high-class photographer", by your definition, one who can produce good results for every single shot would not use editing software. So, yes given that definition, that statement is correct.
The problem I then see is that there are no "high-class photographers" because no person will get every shot perfectly exposed every time, with the exact color balance, sharpness, and contrast that they intended. No one is perfect, and that definition is basically requiring perfection.