The first is that this approach would carry no assurances of success. If just one Democrat, such as Sens. Joe Manchin III (W.Va.) or Joe Donnelly (Ind.), were to cross over on Kavanaugh, the blockade would be rendered meaningless. It might be easier for Democrats to unite against Kavanaugh if they have a Republican on their side, but that wouldn’t be easy for the likes of Manchin, Donnelly, Sen. Heidi Heitkamp (N.D.) or any number of other red-state Democrats who face reelection to do.
The second is that, even if it does succeed in blocking things, the endgame is far from certain. Trump is a proud man who doesn’t often back down and wouldn’t like having his Supreme Court pick held hostage. It’s an open question as to how much he actually cares about tilting the Supreme Court conservative. Yes, it would be a key part of his legacy, but it’s something other Republicans probably want more than he does. If this is a standoff, Trump seems to be unlikely to blink first, knowing how much Republicans want the thing they’d be blocking. (And that goes double if he’s truly afraid of what Mueller might produce.)
And third is the member’s own legacy. Republicans have a finite amount of time to confirm the new justice before they potentially lose their majority in the November election. Whoever would make the decision to take a stand would have to be willing to be the Republican who maybe prevented conservatives from getting a 5-to-4 Supreme Court majority, because there’s a possibility that would be the outcome — however unlikely. That’s not exactly how any Republican wants to be remembered.