I would just like to point out what I believe to be a flaw in your premise. You combine largest, most comprehensive, and best. These are three very different things.

I know some very good local history museums that have 3 rooms, 5000 objects, and survive off of less than $50,000 a year operating budget. They aren't comprehensive, and they aren't large, but they are great at speaking about United States history from a local perspective.

I mean, the Smithsonian Museum of American History is generally considered to be a museum of good quality. They are also relatively comprehensive (as much as any one museum can tell a couple hundred years of history). Due to the relatively visible nature of the Smithsonian Museums though, they occasionally are subject to political pressures (here is an article from the Air and Space Museum of an issue that was pretty well publicized.)

Apart from that, it really is a solid institution, with an excellent reputation, at least among the museum folk that I work with. That being said, the little museum down the street might give you history just as solid.

TL;DR (In this case) Bigger doesn't always mean better.

There is no one single overall best museum. They each have their strengths.

I'm not really an historian; I'm an engineer. For that reason I'd like to put in a plug for The Henry Ford in Dearborn Michigan, where they have an enormous collection of artifacts from the american industrial revolution.