Senators don't win, unless their opponent is also a Senator. The American public generally prefers former/current VPs and Governors.
The exception is Harding, who beat the governor of his own state of Ohio in 1920. But then, in 1920 a dogcatcher running as a Republican would have defeated any Democrat after the aftermath of the Wilson Administration and the bitter aftermath of WWI.As to “current VPs,” a VP is not an executive position but the political heir of the president. Consequently the election of a sitting VP is essentially a third election victory for the sitting POTUS - and you will see a lot of prestige attached to that sitting president - examples are Washington (distorted by the fact that the 12th Amendment wasn’t in place when John Adams became the first VP), Andrew Jackson, and Ronald Reagan. You could sorta throw FDR in there, in that he explicitly won a third term in his own name.
Consequently the VP slot isn’t really a great launching point for a run for the presidency, if you don’t actually inherit the office. Nixon, of course, won the presidency in ’68 after leaving the VP office in ’61. Not sure how many other no-sitting VPs won election to the presidency. Note that Nixon defeated a sitting VP in ’68 . . .