In fact, there is no evidence that the phrase "high Crimes and Misdemeanors" was understood in the 1780s to mean indictable crimes.
On the contrary, with virtually no federal criminal law in place when the Constitution was written in 1787, any such understanding would have been inconceivable. Moreover, on July 20, 1787, Edmund Randolph, Virginia's governor, urged the inclusion of an impeachment power specifically because the "Executive will have great opportunitys of abusing his power." Even more famously, Alexander Hamilton in Federalist 65 defined "high crimes and misdemeanors" as "those offenses which proceed from the misconduct of public men, or, in other words, from the abuse or violation of some public trust."
The logic of impeachment as applied to the presidency is that the president has unique authority conferred by Article II. If he abuses that authority for personal advantage, financial or political, he injures the country as a whole. That is precisely why the framers rejected the idea of relying solely on an election to remove an abusive president from office. Indeed, waiting for the next election is an option that is obviously insufficient when the abuse of power is directed at cheating in that very election.
Justice Joseph Story wrote in 1833 that there are "many" impeachable offenses, none of which is "alluded to in our statute book," because the abuses of power that constitute "political offences" are "of so various and complex a character, so utterly incapable of being defined, or classified, that the task" of enumerating them all through "positive legislation would be impracticable."