But the legislature and executive and the party apparatus are all very different things, with different responsibilities...

That's on paper. In practice, the Preznit says "Congress must act on this urgent issue" and has a Congressperson introduce the legislation that the White House has prepared already.

That's on paper too! From Article II, Section 3 of the U.S. Constitution:

He shall from time to time give to the Congress Information of the State of the Union, and recommend to their Consideration such Measures as he shall judge necessary and expedient;

The annual "State of the Union" speech derives from this, but so does the recommedation of legislation throughout the year.

Incidentally, I had completely forgotten about the next part of that sentence:

he may, on extraordinary Occasions, convene both Houses, or either of them, and in Case of Disagreement between them, with Respect to the Time of Adjournment, he may adjourn them to such Time as he shall think proper;

I don't suppose that this has ever been done.