"The Electors shall meet in their respective States, and vote by Ballot for two Persons, of whom one at least shall not be an Inhabitant of the same State with themselves. And they shall make a List of all the Persons voted for, and of the Number of Votes for each; which List they shall sign and certify, and transmit sealed to the Seat of the Government of the United States, directed to the President of the Senate. The President of the Senate shall, in the Presence of the Senate and House of Representatives, open all the Certificates, and the Votes shall then be counted."
It is a little ambiguous. It doesn't specify precisely that the PotS shall be the one that counts the votes and nor does it specify that the votes have to be accurately counted. However, the current precedent is that the PotS can simply say, more or less, who they want to be POTUS and that's the end of it.
That is the part that Pence fucked up because he would have had authority to count however he wanted, including accepting and rejecting counts from various states.