There are definitely rabbonim who get flooded with calls, even from outside their kehilla/shul. It unreasonable to expect them to be available 24 hours a day to the general public. Even those rabbonim are usually more available for their own tzibur , and will also make themselves available to anyone if the question is pressing. I remember listening to a tape a few years ago (don’t remember if it was Rabbi Reisman or Rabbi Goldwasser) who was called in the middle of night for a pressing question. The rabbi said “I think you’re better off asking your rov, since he is more familiar with the overall situation” and the person answered “I don’t want to bother my rov this late at night”.
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