My thinking is that he is concerned that no matter which way Georgia Senators David Perdue and Kelly Loeffler vote, it could negatively affect their standing in the Georgia run-off election (which as we know is critical to the Republicans holding the Senate).
A motion to contest the election would have no impact on the Georgia runoff. McConnell was referring to the joint meeting of the US House of Representatives and the US Senate on January 6, 2021, which is the last step before the inauguration on January 20. This is one day after the Georgia runoff election.
This joint session is typically just a formality, taking less than half an hour to count all 538 electoral votes. There is a clause in the 1887 Electoral Count Act that allows members of Congress to contest a state's electoral vote. At least one Representative and at least one Senator has to object, in writing, to that state's vote. (The votes are tallied state-by-state, in alphabetical order.) If such an objection does happen, the joint session is suspended, with each house meeting separately to determine how to treat the objection. Each such separate meeting ends with a vote.
Forcing the Senate to take time out of what is normally a short formality would put the Senate in a deep bind. There are certainly some Republicans would would vote to reject the objection, probably a good number of them. The last time this happened was in 2005, where two Democrats objected to Ohio's vote. That one state could have swung the election had the objection gone through. The Senate voted 74 to 1 to reject the objection. The only Democrat to vote for the objection was Barbara Boxer, the Senator who raised the objection. (The House also voted to reject it, but not by that wide of a margin.)
Moreover, unlike 2005, where only one state needed to be nullified or flipped (Ohio was picked for that reason), Congress would need to nullify or flip at least three states that were reasonably close in the 2020 election. Finally, there is zero chance that the House would vote to sustain a Republican objection to any of the states that voted Democratic. It would be a lost cause.