Hi Fiderer, nice to “see” you in here!
Yes my understanding is that if the Senate is evenly split then the VP gets the casting vote in all contests, including that of Majority Leader.
I found this which demonstrates how control can flip back and forth:
“ From January 3 to January 20, 2001, with the Senate divided evenly between the two parties, the Democrats held the majority due to the deciding vote of outgoing Democratic Vice President Al Gore. Senator Thomas A. Daschle served as majority leader at that time. Beginning on January 20, 2001, Republican Vice President Richard Cheney held the deciding vote, giving the majority to the Republicans. Senator Trent Lott resumed his position as majority leader on that date. On May 24, 2001, Senator James Jeffords of Vermont announced his switch from Republican to Independent status, effective June 6, 2001. He announced that he would caucus with the Democrats, giving the party a one-seat advantage and changing control of the Senate back to the Democrats. Thomas A. Daschle again became majority leader on June 6, 2001. Trent Lott announced on December 20, 2002, that he would not continue as Republican leader in the 108th Congress. William Frist was elected Republican leader on December 23, 2002 and began service on January 7, 2003.”