This is a good question. Obama claimed the president has the power to assassinate american citizens without trial and Trump is claiming further powers over immigration, science and the media. None of these things should be the sole purview of the president, this is why rule via executive orders is so dangerous. It would lead to the end of the republic.
The American republic now resembles the late roman one, complete with populist consuls and rioting factions.
This is being exacerbated by the increasing factionalism within the US itself. I'm starting to understand better and better why Washington warned us of political parties in his farwell address
"Without looking forward to an extremity of this kind (which nevertheless ought not to be entirely out of sight) the common and continual mischiefs of the spirit of party are sufficient to make it the interest and the duty of a wise people to discourage and restrain it.
It serves always to distract the public councils and enfeeble the public administration. It agitates the community with ill founded jealousies and false alarms, kindles the animosity of one part against another, foments occasionally riot and insurrection. It opens the door to foreign influence and corruption, which find a facilitated access to the government itself through the channels of party passions. Thus the policy and the will of one country are subjected to the policy and will of another."
Washington's farewell address was a political speech. He was unequivocally a federalist, and his speech was aimed at suppressing anti-federalism. He basically believed there should be only one party - his and Hamilton/Adams' party.
That's why very soon afterwards federalists in Congress passed laws allowing the President to imprison people for reporting truthfully what was happening in Congress (out of the fear it would generate outrage in the populace) and even for mere criticism of the President. For much of his tenure as Vice President under Adams, Jefferson lived in fear of his safety and was even afraid to speak out in private letters, as he believed federalists were reading his mail looking for something to ruin him with. Early America was incredibly factious, with rebellions and armed private militias and mobs perpetrating violence (and threats of violence) on their political opponents.
Politics hasn't changed much over time. Read Washington's speeches like you would Trump's or Obama's.
The letters of most of the major figures of the time are publicly available, as are the newspapers of the era via Google Books and other free, online resources. I highly recommend reading as much as you can. That's a lot of reading, so if you want a shortcut, I'd recommend the book "American Aurora." It is essentially a selection of newspaper articles, Federalist and Democratic-Republican both, and letters, concatenated to form a narrative. You should, however, keep in mind that it is intentionally slanted towards the Democratic-Republicans.
> I feel like Washington is extremely venerated, which makes me a bit suspicious. It's not quite a cult of personality like Stalin but nobody's perfect.
The cult of personality around Washington was quite deliberately engineered and some of his contemporaries - some even nominally his friends - privately lamented that someone (in their eyes) so awful would be, for all ages, venerated as a great man and a hero, but they regarded it as necessary that the nasty infancy of the country would be glossed into something more noble. (I do think his contemporaries judged him too harshly, but things always look different centuries later.)
> This is being exacerbated by the increasing factionalism within the US itself.
The increasing factionalism recently in the US is arguably (barring perhaps some overshoot directly connected to Trump, on both sides) return to the long-term norm as the major political realignments resulting from the New Deal and Civil Rights movements upsetting the pre-existing alignment of the parties completes, and the parties return to a state of cohesiveness.
The American republic now resembles the late roman one, complete with populist consuls and rioting factions.