Very cool idea. The fact that you get to pick you pieces (Alphaminos?) adds a whole other dimension to the standard Tetris format.
It makes me wonder if there is a particular word that would allow one to play indefinitely :)
Incidentally, feel free to check out my own Tetris variant that I made for the most recent jam I was in before this one! https://irishronin.itch.io/not-tetris