A GOP-controlled Senate alone can’t block a Biden electoral vote victory.
There isn’t a presidential election; there are 51 presidential elections (in each state and the District of Columbia). After the electors are certified by the states and D.C., and their votes are forwarded to Congress, those votes will be counted as submitted unless objected to. An objection requires the signature of a representative and a senator. Each house of Congress then meets separately to debate and vote on the objection, and the disputed electoral vote will be counted unless both houses of Congress vote to throw it out. This means if the Democrats control the House and the Republicans control the Senate, and Joe Biden wins 270 electoral votes, a GOP-controlled Senate won’t be able to keep those votes from being counted. How this works was delineated by staff lawyers of the Congressional Research Service, an arm of Congress, in two papers published in 2016 here and here. For a July 2020 CBS News article about the process and examples of what happened in past elections, click here.
The problem you are avoiding is what happens if states electors are unable to meet because the state has been unable to certify the election. This is the real issue that pressed a Supreme Court decision in Florida due to many hanging chads. If it had stayed in the courts then Florida’s electors would not have met, thus Florida would not have cast any votes and Congress would have had to have accepted no votes from Florida. There is precedent for this. There have been times particularly in the early republic when a state did not get their electoral votes in on tie and forfeited those votes. This only matters in a close election. If there is no vote to count then there is no vote for either the Representative or Senator from the state to dispute. Of course results can be disputed and it only takes one Representative of a state and one Senator of a state. And all this occurs before a joint session of Congress.
If no one wins a majority plus one of the votes then there is a contingency election. One in the house and one in the Senate. Of course all of this is contingent that one or more states cannot get their act together to have the electors meet in their state on the proper date and at the proper time. How all this happens is really up in the air as the House has only happened in 1801, 1825 and 1877. If it happens again some or a lot of what happens will be decided by that Congress, and some degree of why it happens may fall on Congress as well. As most likely a state whose electors meet last is likely to still put in the certificates late, giving Congress an opportunity to accept the votes or having to declare the votes forfeit. And the courts may decide hey it is all politics and drop it all into Congress’s lap and not be greatly involved.
Certifying electors at the state level is beyond the scope of this article. Your comment contains several inaccuracies regarding what happens in Congress. An objection requires 1 representative and 1 senator, who needn’t be from the same state. But an objection doesn’t get an electoral vote tossed. As this article explains, it takes majority votes in both houses to sustain an objection and toss an electoral vote. Winning requires a majority, not a “majority plus one,” of electoral votes. If no one gets a majority, the House alone chooses the president; there is not a “contingency election” in the Senate.