Counting electoral votes is not the same as counting the ‘popular’ vote though. The electors were all ratified and sent by all the states. He could only challenge those. The states would just send other Biden voting ones even if he was able to dismiss any.
No one asked or expected Pence to dismiss anything --- just to honor the Constitution and allow the states to review and recertify their electors, as requested --- the states, not Pence.
Counting the popular vote is how the electoral votes are determined. If the popular vote is screwed up, it's quite possible the state's electoral votes are, too.
Prior to Jan 6, legislators from several states wrote to Pence outlining their concerns that the certified votes may be based on election and voter fraud. They requested 10 days to review the certifications. Under the Electors Clause of the Constitution, Pence had the obligation to honor this request and return the certifications to the states for review and re-certification.
Pence, without authority or reason, ignored them.
Keep in mind, January 6 is an arbitrary deadline set by Congress. The only date included in the Constitution is that the President must be sworn in on the January 20 at noon following the election.
This is one of the legal obligations Pence ignored.