Most Americans, like most people, know fragments of songs, rarely able to sing more than a chorus at best, unless they play an instrument. Before TV the computer, pianos were fairly common in the home; now they are rare- so no sheet music in the home.
Composers? More guesses now than correct identifications.

Irving Berlin is better known than most, "Oh, how I hate to get up in the morning" would be well-known, learned in school along with some WW2 history, but not much else.

Berlin died almost 25 years ago, so is little known to the 30 and under set.

He was still on the hit parade in 1935-1945, when I was in my teens, so I would know a number of his songs, including "White Christmas," bought as a record, often heard on the radio, and possibly picked up as sheet music.
Now "W. C." is played in the Christmas season, and the average person cannot name the composer, any more than they can name the composer of "Rudolph, the Red-Nosed Reindeer."