They took their inspiration from the theory of 'mixed constitution' in ancient Greek philosophy:
https://www.constitution.org/rom/polybius6.htm
The executive office was supposed to be powerful, but not so powerful that it could act without the support of the legislature branch and the electorate. Everything was supposed to be balanced so that none of these three branches could function without the other two.
I should think that what threw balance apart - and the philosophers always acknowledged it to be a delicate balance, easily thrown off if you aren't careful - is the formation of political parties and the loyalty of people in the legislative and electoral branches to parties. For the mixed constitution to work, every stakeholder's highest loyalty should be owed to the republic, and their every action should be for the good of the republic.
But when you have parties and factions that are for the promotion of certain special interests instead of for the common good, that's exactly the sort of thing that upsets the republican state. As Aristotle wrote in
Politics, "Tyranny is monarchy ruling in the interest of the monarch; oligarchy, government in the interest of the rich; democracy, government in the interest of the poor; and none of these forms govern with regard to the profit of the community."
Instead of senators and voters holding an unfit or tyrannical President or other magistrate to account, they form ranks behind him if he happens to represent the party they belong to. Without the party machine supporting them, could Trump, Clinton, or Biden have gotten as far as they did? In a healthy republic only somebody with real grassroots support as an individual would amass the clout required to win elections.
The beauty of the mixed constitution is that no one however great they might be can exert any political force without the cooperation of the other two branches. A President serves at the pleasure of his senate and electorate; if he fails to keep them onside, the senate can obstruct his policy and the people can see to it that he is put on trial once his term expires. And it is much easier to impeach an individual who takes liberties while holding office than it is somebody who is backed by a party machine. Parties and factions distort the balance because they take away the effectiveness of the other two branches in holding a President to account.
Factionalism was also the cause for the ruin of many great republican states in the ancient world. Carthage's republic became paralysed by the struggle between Hanno's aristocratic faction and the popular faction led by the Barcid family. The Achaean League in Greece was destroyed by the conflict between the pro-debt relief faction and the oligarchic faction that was opposed to property redistribution and debt relief laws. Many other Greek cities and federations destroyed themselves through vicious factional conflicts involving foreign policy. There were pro-Macedonian factions who fought with pro-Roman factions in the Greek cities, while the Macedonians and Romans themselves fought colonial wars with each other over the possession of Greece. And of course the most famous example of factionalism destroying a republic is the struggle between the popular and aristocratic factions in Rome, leading to the end of the republic and a return to monarchy.