In addition to the issues already covered in Chipster's answer, there are a few more reasons not to do this:
Not Much Time
The window for the confirmation would be very narrow. While it's about 2.5 months from the election until the Presidential inauguration, the new Congress convenes on Jan 3, only 2 months after the election. And there are typically significant recesses during that time, due to the holiday season.
There is enough time for someone to be nominated and confirmed on an expedited basis, but it leaves little room for error anything delays the nomination. For example, if 11th-hour accusations are brought against the candidate, they would want time to investigate them before holding a vote.
If something came to light during the hearings that caused the Senate to decide not to confirm the nominee, then they would be on an insanely tight schedule for the President to nominate someone else and then the Senate run through the entire vetting and confirmation process again, and it would be in the middle of the holidays when the Senators would highly prefer to be at home with their families.
Most Senators Want To Get Re-Elected Next Time, Too
You say voters have a short memory, but what has been the single-largest point of debate surrounding the appointment of Justice Barrett? It's been reactions to what happened 4 years ago when the Senate refused to confirm Garland. It's bad enough to go back on promises you made 4 years ago when you were making a promise about a hypothetical situation. It's a lot worse to go back on a promise that you made 4 weeks ago about this exact particular case, obviously intending to violate it from the beginning.
The reaction to that on an election 2 years later won't be as bad as it would be on an election tomorrow, but it will definitely not be zero. Voters' memories may be short, but the Internet's memory isn't and neither is the other party's, who will surely play videos of you making that statement on repeat on their campaign ads the next time around.
Only 1/3 Of The Senate Is Up For Election
During each campaign cycle, only 1/3 of the Senate seats are up for election. Thus, the Senators who will be taking a massive hit on this at the polls in 2 years are not the same ones who are under political pressure at the polls now. Regardless of how willing politicians may be to brazenly lie to their constituents, very few of them will be willing to sacrifice their own political career in order to reduce political pressure on some of their colleagues.
Holding The Vote Has Positive (To Them) Consequences On The Current Elections, Too
While it's true that some GOP Senators may be facing additional pressure at the polls because of the vote, others will likely experience a benefit from it. It's hard to understate what replacing Ruth Bader Ginsburg with a conservative justice means to the conservative voting base. There are few things that will energize the conservative voting base more than that.
Also, this affects not only the Senators, but also for the President who still wields considerable influence over Senators and who is himself facing a tough re-election bid. The conservative base will absolutely see this as a large accomplishment for President Trump. And then those additional people showing up to vote for Trump and perhaps also a GOP Senator will also be likely to continue ticking boxes next to GOP candidates all the way down the rest of the ballot.