Show me in the statute where anyone other than the VP has the duty.
Both the 12th Amendment and 3 U.S. Code ยง 15 say that the President of the Senate will oversee the proceedings. Under Senate rules the Vice President is the President of the Senate when present. In his absence then the president pro tem is the President of the Senate. In the absence of both then whoever the president pro tem designates is the President of the Senate. Nothing says that the vice president has to be there.