The GOP wanted to confirm a justice well before the midterms to give campaigning members a "W" they could use on the campaign trail.
First they made the decision to limit the release of documents, hoping that would give Dems less ammunition. That strategy largely failed
When Ford's allegation came out they tried to push the vote through, saying it was the "11th hour" and too late to hold any relevant hearings. That largely failed, and they had to hold a hearing
Sensing how bad the optics were to have 11 old white men questioning an alleged sexual assault survivor, they hired an outside council (female) to ask questions. She failed to make Dr. Ford less credible, and the GOP senators grew tired of staying silent during Kavanaugh's testimony and came out with effusive praise for someone who had just been accused of being sexual assault and attempted rape. The hope appears to be that putting him on the court is worth the risk of looking like they are supporting a sex offender. Even the kindest reading shows it didn't go particularly well for Kavanaugh.
Now here we are: Flake and Murkowski are calling for
a limited 1 week FBI investigation. The GOP is weighing the potential risk and reward. If they deny any investigation they risk losing the vote, and it will be horrific if new accusers (perhaps emboldened by Ford's gripping testimony) come forward in the days to come. If they push for an investigation they are in an equally tough spot - the best case scnerio is that nothing new emerges, whereas it would be catastrophic if either i) others corroborate parts of Ford's story or if ii) new allegations emerge. Then they are left with the following choice: dump him or approve. Dumping him then would be the long-term play, but likely infuriate their base just weeks before the midterm. So I'll imagine they'd continue to make the bad choice, and vote, and put every Senator on record.
Avenati is just itching to toss more gasoline on the fire, and every single former classmate, girlfriend and buddy of Kavanaugh just heard him talk about how he was a model student, a virgin and had never blacked out in his life. It seems extremely likely that at least a few might say "hey, not at all how I remember him". More than a few probably have some yearbooks, polaroids (remember those?) or letters stashed away in memory boxes. How many of them will spend this weekend digging through them, looking for anything with his name or image on it?
tl;dr - the GOP continues to play a very dangerous PR game. With each step they are betting that nothing else comes out, and nothing more can be corroborated.