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Yes I prefer to deal online when I find myself dealing with an industry where the live human beings only work a 9-5 schedule. I just checked and all Geico agents in my area only work about a 9-5 schedule. Since I work myself during those hours (and sometimes a lot longer hours) yes it is difficult for me to get to a live agent. I'll deal with folks by phone or in person if absolutely necessary--and perhaps it will turn out to be absolutely necessary in this case--but it is never going to be my first choice and providing good online service--along with flexible office hours when I do need to talk to a live human being--is going to be a factor in my choice of company.
Most consumer facing businesses, recognizing that a significant segment of the market themselves have jobs and aren't available 9-5, are open longer hours. For example the dealership where I bought my car is open until 8pm and their service department is open to drop off my car when maintenance is needed starting at 7am. I enjoy dealing with live human beings when they offer hours that are friendly to my own schedule and--if we are talking about the telephone--they are reachable with a minimum of time on hold. The Geico agents in my area don't offer flexible enough hours.
Perhaps you could talk your employer into changing your work schedule to the graveyard shift, that way it wouldn't be so inconvenient to handle your business during your personal time.