The VP, while President of the Senate, is neither minority nor majority leader and only has a vote when the chamber is equally divided.
There isn’t a Canadian equivalent, but the closest one would be our Speaker of the House, who has about three orders of magnitude less clout than the U.S. speaker of the House, whose legislative power is the closest U.S. equivalent to the Prime Minister (legislative power only), albeit close to two orders of magnitude less clout.
I fully understand. IF the Senator from Alaska had voted on all the Amendments being offered and voted on in the past 24 hours. Presumable He would be voting YES which would make the vote 50/50 thus requiring VP Kamala Harris as the VP to cast the deciding vote and as such would very likely side with her boss and then the administration owns the results of killing those Amendments.