Seems that Senate Majority Leader Mitch McConnell's opposition to a president nominating any person of his or her choice to the Supreme Court is a newish thing. Because back in 1970, writing for a law review, McConnell was all about the proper constitutional roles for the president and the Senate when it comes to the Supreme Court.
At the outset, the Senate should discount the philosophy of the nominee. […] Political and philosophical considerations were often a factor in the nineteenth century […] but this in not proper and tends to degrade the Court and dilute the constitutionally proper authority of the Executive in this area. The President is presumably elected by the people to carry out a program and altering the ideological directions of the Supreme Court would seem to be a perfectly legitimate part of a Presidential platform. […] The proper role of the Senate is to advise and consent to the particular nomination.
Of course, it could be that McConnell reserves these constitutional rights just for Republican presidents. Because he sure has changed his tune in the intervening decades.