The President can nominate the SCOTUS judges, however, as we have seen with many nominations in the past, ideologically pure candidates don't normally make it through the Senate confirmation process.

If the President is handed a Congress with a narrow majority, or a minority in the upper house, then they have to moderate their choices for the court.

As several nominees who have been appointed to the court have shown, (Souter, Breyer) even those chosen by a President who has a majority in the Senate don't often stay on the ideological reservation.

The President could nominate Herr Schickelgruber to the SCOTUS, but that doesn't mean they're ever going to sit on the bench.,