OK I went and gave it a listen. The copy I have is in this 8 disk box.

1. I like this piece very much.
2. No idea if I like this more or less than any other Beethoven String Quartet. I like them all very much.
3. I swear I heard at least ten distinct samples Rodgers & Hammerstein Sound of Music soundtrack.
4. I was so convinced of this I was expecting to get real red meat when I googled on the following terms: (rodgers hammerstein sound music beethoven string quartet 13). Alas, all I got was a long list of orgs who had both of those items in their immense repertoires, but nothing like grouchy musicologist's friends writing back and forth pro and con at length on similarity and difference.
5. A conjecture. My mom's favorite record was the Sound of Music soundtrack, and she had simple taste. I bet she would have liked the "grosse fugue" on one listen, from which I would argue that this piece is accessible.

(Also Rodgers and Hammerstein were going for a German folk music sound, so perhaps Beethoven and they were both independently derivative of the same sources. Or this connection could purely be a figment of my imagination.)