Well, technically we're still singing just the Chorus from the 1892 hit. The verses (three of them) not so much.
You can hear the verses by going here, and clicking on "Listen":
Bicycle Built For Two [musicnotes.com]
This is pretty common with older songs--only the chorus gets remembered, and the verses get lost. Sometimes it's really weird: in "The Band Played On" (Casey would waltz with the strawberry blonde), the verses are not only in a different key, they're in a different time signature!
And you never find out why her brain was so loaded it nearly exploded.
At least with "Daisy," the verses make sense relative to the chorus.