So can any type of file -- that doesn't have any relevance to the supposed design of every file type in existence. Now, later versions of PDF do have explicit support for signatures, but what does this have to do with preventing OCR? OCR reads a file, it doesn't change the original file.

Can't one just remove the signature and re-sign it with anything else after tampering? Who verifies PDFs that hard?

PDFs are merely an collection of objects, that can be plainly read by reading the file -- some of those are straight up plain text that doesn't even need to be OCR'd, it can be simply extracted. It is also possible to embed image objects in PDFs, (this is common for scanned files) which might be what you are thinking of. But this is not a design feature of PDF, but rather the output format of a scanner: an image. Editing PDFs is a simple matter of simply editing a file, which you can do plainly as you would any other.