It's not that he overgeneralizes; that is endemic among philosophers, and is probably part of the job description. The same may be said for applying well-reasoned ideas to inappropriate subjects; it's expected.
It's that he philosophizes at all.
The reason Popper is not more widely appreciated is, almost everyone his basic reasoning would appeal to is too busy, getting stuff done, to spend time reading philosophy. Which is why we have a civilization capable of supporting philosophers; there's a lot of stuff that needs doing, to make life worth living for as many people as possible.
The problem with this is that Popper, whose work has inspired no inventions or innovations that I am aware of, would have done far better to apply his principles rather than continue to refine them. Had he devoted his energy to, say, driving a cement truck, and talking about his ideas with the people whose driveways he poured, he would have influenced public thought on a more fundamental level; and while he would not have been as well-known for his philosophy as he is today, his philosophy would be far more widely known, understood, and accepted.