Well, of course, that’s more-or-less why the US has the system that it does. The President is analogous to the king - head of state, not elected by or accountable to Congress, able to appoint the heads of government agencies, but not able to get much done unless he can persuade Congress to pass the legislation he wants, and to accept his appointments to great offices of state. The big change was to make the president elected by the electoral college, and holding office for a limited term.
Because the king didn’t have that mandate, as democratic ideas took hold in Europe european monarchies evolved, with more and more real power passing to prime ministers/chancellors/what you will who were more and more dependent on parliamentary support, and less and less residual power in the hands of the monarch. This happened at different rates in different countries, of course, but the overall trend towards monarchs becoming more and more constitutional, or simply being abolished in favour of more-or-less ceremonial presidencies, is pretty clear.
The US, as an early adopter, gave their head of state/government the authority of a a democratic mandate early on, and so he was able to resist the trend towards making the executive accountable to a parliament. (That, plus the early adoption of a written constitution which crystallised the relationship, and which is difficult to change.) Hence the “imperial presidency” - the US president still having powers and a role broadly analogous to those of an 18th century British monarch.