I am pretty sure it would. I remember reading a scenario once that a player on a punt team taps the ball, a player on the return team picks it up and runs for about 15 yards and fumbles it, and the punt team recovers. The ruling was that the return team had the option of taking the ball at the spot the player on the punt team first touched it. The key is the punt team must have control of the ball. Not sure if my verbiage is exact, but you get the gist of it; that you are correct. Pretty sure I am missing something about the punt team touching it first being some type of illegal touching, but no flag is thrown.
Similar scenario in a game I was an assistant in. We punt, cover it well, and our kids lazily slap at the ball and guys start running off the field. I see one guy running toward the ball, and I am saying in the headset 'that's a live ball' over and over and over and over again. the return team scored a TD. When coaches asked if I was sure, I said YEP, and then I asked, did any of you hear a whistle?