"executive and legislative branches are one and the same"
That was the claim. I know in theory it is different, but also in reality here in germany I remember a scandal, where a minister of transport stopped a rule breaking other driver - big problem for the minister, because he was not allowed to do so (we have nice german word of "Amtsanmaßung" for this). So that sounds pretty separated to me.
But you have a head of state who can, in theory, remove the head of government at their own discretion, right? In the US the head of state and the head of government are the same person.
Yes in theory, the King Charles III and/or his representative the Governor General, could remove the Prime Minister, though the Governor General is chosen following advice (i.e. choice) of the Prime Minister, so I can't imagine that happening in practice.
The US president is head of the executive branch, one co-equal third of the entire government, also including the courts (article III of the constitution), and congress (article I).
If you have the majority, you can’t even be removed through a vote of no confidence.