I agree with Bill D., Mountain Tyme, and Frank H. that many "2-chord song" can be done with more, if you're inclined. I make a distinction, though, (which is not always hard and fast) between songs that can get by with only 2 chords and songs that really only need two chords.
Lots of songs really sound better with added 7ths, minors, minor 7th and 4ths, even though they could be sort of played with 2 chords.

Songs I play with only 2 chords and really don't feel need any more include:
Sinner Man
Drunken Sailor
I Ride An Old Paint
Fiddler On The Roof (Theme song) [Sheet music gives 3, but I see no need for anything but E and F]
Alouette
She Moved Through the Fair
Rocka-my Soul In the Bosom of Abraham
Skip To My Lou
Go Tell Aunt Rhody

"2-chord songs" that I really think benefit from added chords include:
Mary Had A Little Lamb
Shalom Chaverim" [I use Am, Dm, and E(7)]
Hey Ho Nobody Home ( I use Am, Dm, and G)
Tom Dooley (adding a 7th helps)
900 Miles
Doggie In the Window (I use G, Em, D and D7)
Down In The Valley (G, D, D7)
Twinkle, Twinkle, Little Star [C, C7, F, G(7)]
Jambalaya (I guess I only use G, D, D7, but I 'cheat' by playing melody licks that deviate a bit from the basic chord pattern)
Pasture of Plenty is similar. I think I use 3 chords (don't have a guitar handy now, so I'm reconstructing from finger memory), but I do various runs that add notes to those chords.

Rounds often use simplified chords so they can be sung that way. If you play the natural or obvious chord progression to some songs, they won't work as rounds, but if you do them a capella, they work fine. Same thing for "partner songs." A case in point:
"Hard Times Come Again No More" can be sung as a partner song to "Cottonfields," if done a capella or with the appropriate 3 chords. On the other hand, if you add some of the supplemental chords to "Hard Times," it screws up "Cottonfields."

In other words, if a song played with 2 chords sounds like it's being played by a beginning guitar student, I don't think it's essentially a 2-chord song. But there are some songs for which even good guitarists use only two chords.

Arge