I think it is broadly the case.
Way back, I got to follow around one of our VPs for a day. It was exhausting, meeting after meeting, people talking to him. I realized a VP is really more of a cheerleader than decision maker. People present him stuff where only one decision is plausible, they leave out anything contrary to that for the most part. So, he pats them on the back and says "Great job, carry on" and they leave, all happy.
I really think this is true for every President, maybe he sort of sets out some broad policy objectives, at times, and relies on his underlings to go execute them, but even his broad policies can be massaged by the underlings at times. They nominate over 4000 appointees, obviously he would know of maybe 50 of them, the rest are chosen for him. He says he wants to cut taxes, well, folks in Congress come up with the plan, really staffers in Congress, and get something passed, usually with all sorts of special interest items included.
This stuff is, in my view, WAY to complicated for any single individual to have more than a passing influence over some large item in a bill.