The new wave of authoritarianism uses the bureaucracy and state machine against itself. They've learnt from the past, there's no Night of the Long Knives, or the burning of the Reichstag, that's too overt.
Look at Orbán, Erdogan, and similar new authoritarians, the coup is a slow grinding of every facet of democracy: grind the press, grind the judiciary, throw a new law into the mix made up by your cronies in the parliament/Congress, keep grinding at the foundations of democracy over decades and you get a shell of institutions that used to be the foundation of a democratic state being re-purposed to push the agenda of the dictator.
The veneer of institutions is the point, it's to make you have exactly this reaction:
> I genuinely don't understand how this can be a coup when all three branches agree.
Because you are expecting an overt coup, not a slow moving covert one.