Perhaps presidents are mainly there to make speeches and slightly tweek a few policy decisions important to constituents, but government itself is pretty large and goes right on between administrations. (Which isn't always a bad thing either).
The actual "government", in the sense of who determines how life actually goes on in America, is distributed across a web of key employees in major corporations and governments of all levels. These are not necessarily the CEO (who is often the shareholder's & employee's punching bag); rather, it's the anonymous guy who has been there his whole life, has a huge network of connections within the company, has developed a large degree of expertise in the business, and holds the trust of both the nominal decision-makers and the line-workers who do the work.
Witness how we've had 18 government shutdowns since 1976 [1], and life generally goes on during them. (Interestingly, most of them were during the Carter/Reagan years, lest people think government has gotten more dysfunctional recently.) Also observe how the President usually gets blamed or credited for the condition of the economy, which he really has very little control over. People want a figurehead that they can point fingers at; it's usually not the guy who actually holds the reins.
[1] https://en.wikipedia.org/wiki/Government_shutdown_in_the_Uni...
But let's also step back and recognize that "the president" has always been at least the president himself and his adminstration. Of course the network expands from there and the key employees that you mention in important companies include individuals capable of weilding considerable 'asymmetric' power.
Lately, I can't help but think of Cigarette Smoking Man.