If a senator had the advantage and would win with the existing count, how would a batch of single-vote-for-president ballots hurt his situation?
There wouldn’t be any votes against the senator on those single-vote-for-president ballots.
there are other advantages to a single candidate vote ballot, and those are that it is easy to describe and allows fast execution, permitting in theory the generation of more accurately targeted ballots and also permitting the generation and insertion of more ballots into the count.