The US constitution intended the president to be non-partisan. Why did the founders not write some controls and checks to prevent the president from being in a league with a political party?
The President is head of state and head of government at the same time. To function effectively, a government needs the ability to get laws and especially budgets passed (power of the purse), which is generally reserved for the legislature. So the President needs to build working majorities in Congress, and that makes him or her partisan if Congress is partisan.
In my opinion one of the fundamental weaknesses of the United States Constitution is that it doesn't acknowledge parties and regulate their role. Some people might call that a feature and not a bug, but electorates have a tendency to form parties -- or "very stable factions" if they don't adopt the trappings of a party.
Seems to me there is an implicit assumption here that the very notion of political parties is bad.
That's not an obvious conclusion to me:
People will usually congregate with like-minded people in many circumstances. In the sphere of politics that will start to look like a "party". How would you go around abolishing that behavior?
A party is a useful shorthand for "stuff I like/stuff I dislike". If I am mostly concerned about business efficiency, then a center-right party is likely to work for me. If I am very pro workers rights, then a center left party will do. If I do put in the effort of figuring out parties' general orientations, then I can be lazy in the future and make some assumptions (recent question about that was interesting) as to how they probably align with new interests.
An alternative is to run everything on single issue votes, by direct democracy and referendums. That's more attractive in theory than it is is in practice, at least to me. Things like budgetary problems from California's Prop 13 or Switzerland's really belated grant of full women's right to vote make me quite cynical on direct democracy.
These are the good parts of parties, when they work correctly. At some point you have to design a system making some assumptions about likely conditions, and even some possible dysfunctional ones, which is what the founders did.
But it seems unfair to blame some shortcomings of the founders for the dysfunctional behavior of today's political establishment and voters.
People like Michael Moore or Ann Coulter make a living peddling untruths and raising hatred against "the others".
Republicans engage in rather obvious voter suppression shenanigans in Texas and Democrats are unable to stop contemplating packing the court, a blatant attack on the US political system.
Members of the public frequently say that "all politicians lie and they are all corrupt" and are not challenged.
The current problems are caused by both parties' incapacity to muzzle or ostracize their extreme wings and a growing tendency by the public to demonize anyone on the opposite side. Rather than complaining overmuch about what the evil Reps are up to (if you are a Dem), maybe a well-placed "will you shut up, man", to a fellow Dem that advocates something stupid, like fully-open borders, would be a good thing. Flip the argument if you are a Rep hearing one of your fellow Reps whining against masks.
Is there a design problem?
In the past, I thought the US had a unique capacity to bring in fresh political ideas by its primary system. Bill Clinton or Ronald Reagan were relative unknowns by the standards of most democracies. Before Mitterand, in France, won on his 4th election, he had contested 3 others on 7 yrs cycles. He was old and so were his ideas.
Nowadays that looks like a design flaw instead. US primaries reward extreme positions decided upon by a small minority of uncompromising voters. Trump's "qualities" are well-known by now, but Ted Cruz, the last Rep to drop out is on the record as being against abortion even in the case of rape. Moderate Republicans got turfed out early on.
Bernie Sanders, a man who wasn't even a Democrat for a very long time, and one who holds views to the left of many a European center-left party got a huge amount of traction in 2016 and 2020, again by appealing to a minority of very committed voters. His viability in a POTUS election, at least against any functional Rep candidate, seems very doubtful to me given the general US electorate's position on taxes and social support systems.
So, in that sense, there is a design flaw, but not so much in the notion of party-affiliated Presidents, but rather in the way the parties choose their candidates.
The US President is elected by way of a party, because it offers the best chance for him/her to get elected because of party support (the entire party would likely vote for him/her in the general election) and the party that backs him/her up to push forward their agenda.
It's basically a win-win situation for both the president and the party, and so this system is stable (in a Nash equilibrium). It works so well that virtually no change is necessary for the system for the past 3 centuries.