This isn't entirely correct. The Lt. Gov. can vote to break a tie vote in the Va. Senate, just like the VP can break a tie in the U.S. Senate. Lt. Gov. Chuck Robb broke a tie vote on a very controversial issue that came up in the late 70’s or early 80’s. I vaguely recall that the bill provided funding for abortions in the case of incest or rape. I remember the vote both because of the underlying controversy over the bill but also because the tie-breaking opportunity was rare.
You are correct; however, as you point out, such occasions are so rare as to hardly matter. I am not aware of Kaine EVER being in such a position.