The purpose is to discourage anyone who opposes them from accepting the nomination, lest their family gets dragged through the same thing.
Partly true, but also they need to find a way to force him to recuse from the “sex” cases before the court, specifically the trans case. If they can get him recused “because he is under a cloud of impeachment” then they must know they have the four votes to uphold the lower court/EEOC decisions causing a tie at the USSC, which means that the lower court decisions win, (just as happened with the 5th circuits DAPA decision).