Why did the FAA ban all supersonic flights over the United States, rather than only supersonic flight in densely-populated areas?
The short answer is a bad PR fallout following the Oklahoma City sonic boom tests, even though the tests were generally positive.

[The National Opinion Research Center] reported that 73% of subjects in the study said that they could live indefinitely with eight sonic booms per day.
(...)
The FAA's poor handling of claims and its payout of only $123,000 led to a class action lawsuit against the U.S. government. On March 8, 1969, the government lost its appeal. The negative publicity associated with the tests partially influenced the 1971 cancellation of the Boeing 2707 project and led to the United States' complete withdrawal from SST design.

The tested 96 Pa (2 psf) overpressure -- what an SST produces and is far lower than military jets -- according to concordesst.com is:

(...) equivalent to the pressure one would feel when putting a hand out of the window of a car travelling at 30mph.

A 1973 article by The New York Times, &quot;Supersonic Civilian Flights Over U.S. Are Outlawed&quot;, does not give reasons, only that it was expected.

In a new order, the Federal Aviation Administration virtually outlawed today supersonic, flight over the United States by civilian aircraft, effective April 27.
The rule had been expected for three years, since the F.A.A. first proposed it in April, 1970, in a notice inviting comment for or against the action.
Accordingly, it came as no surprise to manufacturers of supersonic transport planes such as the British‐French Concorde and the Soviet TU‐144.
(...)
Congress canceled the Boeing‐General Electric program for an American SST twe years ago, for budgetary reasons.

The New York Times also reported in 1970 about the supersonic trials over populated areas in &quot;West Scotland, Wales and England&quot; with no reports of any damage.
Open questions
Was it politically motivated? Were there really no solutions such as the proposed corridors? Those and @CptReynolds' commentary that [the US] banned it to &quot;prevent a European aircraft from capturing 100% [of the SST] market share with US airlines&quot; are worthy of posts on Skeptics or Politics.SE. Over there they'd know where to dig deeper.
One of the things I found for example was a ban on Concorde before it set a tire in New York. So it was a ban that came before any evaluation for its subsonic noise. The Supreme Court eventually intervened and lifted the ban (The New York Times, 1977).
Another find was a quotation from Ohio Senator Clarence J Brown, saying:

I see no reason to oppose the SST [meaning Concorde] just because we were stupid enough to get out of the SST business ourselves (concordesst.com).