The Electoral College has decided that the second-place candidate for president should not be the vice president. It has allowed this “running mate” system to develop to avoid the problem of the vice president and the president being at odds because they usually are on different end of the political spectrum. The vice president does not have much power, but having Hillary Clinton in charge of the Senate in its present composition would be a good check and balance. (Another question is why the leader of the Senate is someone other than the vice president.)
Since the person who is vice president is not vice president for any reason other than his association with the president, if the president is removed from office, then the vice president should be also.