But POTUS doesn't draft it, and never really sees it until it passes both houses and ends up on his desk.

I’m going to quibble with that a little.

Lobbyists draft legislation all the time. I’m sure that Presidential staff does so as well.

But in either case, it takes a member of Congress to introduce that draft legislation as a bill. And then the houses of Congress amend that bill to their heart’s content.

As to seeing the bill, once introduced by a member of Congress, it becomes public. Anyone can see it - even easier today as it’s available on the internet. I’m quite sure that every President has staff that read the bills in detail and prepares briefs for the President. And every President spends time on the phone or in person with members of Congress to try to influence them to deal with legislation in the way he would prefer.

The President can be as active or as passive as he chooses to be in the legislative process. But in the end, it is Congress that must create the law and not the President.

—Peter