I suspect that the job of president is almost entirely reaction.
Sure everyone comes in with an agenda and works on getting it working – but everyday your day starts by reviewing all the shit that happened all over the world while you were sleeping. Turkey and Greece posturing toward each other-- requires a Presidential response. A tidal wave in Japan or anyplace else – Presidential response. Florida has a few days of freezing weather and frozen concentrated orange juice is in short supply all around the world – presidential response.
So after getting the diplomatic corp, the military, FEMA, and NGOs like the Red Cross to address all the man made and natural disasters that have happened anywhere, you can look back toward the fights that congress is insisting upon on the Hill, what the courts are going to, or have already struck down that needs your attention. After that you can see what Administrative Branch employee said something stupid, or took a bribe, or quit to run for office after neglecting his or her responsibilities for the last month.
Then you have to spend a few hours in meetings that have been scheduled before disaster struck. Worry about interest rates, hear about grazing cattle on Federal Land, have some policy wonk drone on about international tariffs for an hour – then make a tough decision that is going to piss off an ally or an enemy, AND half the American population. Have some lunch before farmers and truckers and construction workers (well their representatives) tell you they need more pay and better working conditions and while they file out of your office the industries that employ all those people file in to tell you they cannot afford to give those people the raises or better working conditions they require. and absolutely everyone at every meeting has a good point, but also has an agenda that goes five steps beyond the actual truth where they have the moral high ground.
Absolutely everything the POTUS does has big and important consequences, often effecting the future of millions of citizens and sometimes involving life and death. Giving speeches and rewarding those who go above and beyond are the easiest parts of the job. And as others have pointed out- those things matter too because you have to get public support (and even then the opposition shits all over your sincere efforts- e.g.: right to choose issues and universal background checks for firearm purchases are hugely popular with a huge majority of the voting public yet remain bitter battlegrounds)
On top of this, you have to work with people who oppose you at every turn, people you hate, and you cannot fire most of them. You have to notify the top leaders from both parties to take any significant steps in many areas (maybe most areas??) You are constantly forced to make decisions without having enough information – and an hour or a day later information may and often does surface that makes the sincere decision you made seem either silly or stupid.
Does anyone really think Jon Stewart wants to wake up every morning and find out how many civilians were killed in Ukraine? Or what violence erupted in the West Bank? Or how some country defaulted on their loan and now international markets are going to be in the toilet for day or two? That is why I say give him an appointed job where his popularity and poll numbers are meaningless and he can focus on advising the administration on how to best react to an event, or how to use the country’s resources to solve a longstanding problem, or which path forward will net the best result. Hell, he might be a good press secretary, but I think more behind the scenes myself.
And one last thing that needs to be said in my view. Trump was never the president in any real way except ceremonially during speeches and state dinners and as a figurehead. Everything he personally touched turned to shit, there is no evidence he ever digested one single security briefing during his entire administration. Everything he delegated (especially to Jared, but to all those he brought in with him) were dropped ten minutes later. Besides stirring up shit at rally’s his biggest accomplishments were binders full of unrelated ideas upon a topic like healthcare or national defense. Every institution in the Administrative Branch was weakened and some were nearly dismantled. Trump was president like anyone with a fantasy football team is an NFL coach.
For the right person Stewart might be a totally kickass chief of staff. I could totally see him going to the hill and belittling some obstinate senator into complying just to stop him from ridiculing the office holder in front of his or her staff. His bullshit meter is finely calibrated and he might be able get results just by pointing out to an obstructionist how stupid they are going to look on the late night talk shows that night or next week. He is often so completely brutal and honest that some people who really care about how they are polling might respond to : “Do you mean to tell me that you would rather . . . . .”
