Ok - an update for my own post. It appears they *are* soldered normally. Mine must have been just barely soldered at all and completely disappeared when it broke free.

I fixed my player by putting a small amount of solder on each contact spot on the board. Then putting the button over it.. then carefully heating the little metal wing that covered the soldered area. After a brief second, the solder softened up and covered the wing enough that the button was soldered in place and now works perfectly. It takes a steady hand and a good eye - but it works.