I think that's overstated.
First of all, it was the Magna Carta which established that the king was not above the law, so in that sense the founding of the United States was more "and we don't need a king".
Second, there are frequently legal discussions in the present day as to whether a sitting president can be charged with a crime, so there is at least a substantial idea that there is some "sovereign immunity" (quotes because that's not what sovereign immunity refers to)
Third, the founders were also well aware of the threat of political motives for prosecutions and wanted to diminish them with various balance of power checks and balances.