For every location on earth, there are parts of the sky that the sun can never be in (e.g., directly overhead anywhere outside the tropics).
I think the sun can be in about 40% of the sky (a hemisphere).
I'll withhold my reasoning/calculations for now, because I'd like the opinion(s) of others more-expert (most are) than me in this subject.
Please weigh in. I believe this "simple" geometric question will have been answered centuries ago, but I have not found it as an answer, nor even as a subject.