This sounds really fun!
One suggestion though, keep all the letters, but lump all the rare letters like J, X, Y, Z, and any other that you identify as rare, together. Then hold a draw after the sign up is full and randomly assign participants their letters, with the rare letters group being treated as one.
This way, you do not need to discard any letters before-hand and can just rely on the luck of the draw to select the 16 letters (or 15 distinct letters and 1 rare letters group) that will feature in the game proper.
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