It's not the President's style, in fact, I mostly admire him for his ability to keep his eye on the ball. (And that ball for the moment is managing what's going on now...an ongoing task filled with public relation land mines.)
In my opinion you have it right, it's the Republican Congress who should do this lifting. The Senators in particular, who owe their majority status to the man in the White House, shouldn't have required days to come to the defense of the President's honor and character.
Yep, less sniping at their own, more defending the agenda is what is needed. Especially from the "conservative" punditry and the "Republican" Senators.