Separation of powers is a common justification for this state of affairs but it does not really explain anything as such. Another way to look at it is that the US constitution is really old. The fact is that the idea was very popular at the time the US constitution was drafted and apparently influenced it greatly.
And the US was charting new territory, there wasn't much past experience to look back on. Even the very idea of a written constitution in the modern sense was kind of new (there were a few like Corsica but most European states didn't have one).
For most of the nineteenth century, the main issue with this constitution was not its presidential nature or its effectiveness in enacting legislation but the question of “states' rights”. For the most part, states' rights advocates were defeated and the constitution took such a symbolic value that tinkering with it or drafting a new one like many countries have done during the last century now seems all but impossible. Plus the US did not go through any major defeat, foreign occupation or authoritarian regime of the kind that prompted thorough changes in several European countries.
But if you read Montesquieu, separation of powers was not necessarily meant as rigidly as it was implemented in the US. The practical difficulties are numerous and modern constitutions tend to take a more nuanced approach (hence the apparent peculiarity of the US system compared to much of what came later). More recent constitutions also tend to be (much) longer than the US constitution.
In the US, it does kind of work however because the lower levels of government and the courts have a bigger role than in many other countries, including England (I am not writing “the UK” because the institutions of Scotland and Northern Ireland also have extensive powers).