Quick drop isn't a standard feature in the Tsu ruleset. You can enable it, but it's designed to keep the pace of the game more about building combos and keeping you aware of what your opponent is up to. Other Puyo modes have it as well, but Tsu versus never has.
It's there, but you can blame TTC for that. Seems to be a bit improved from Tetris Party/Axis, but it's been ages since I've played that. No infinite spin either.
Thanks to the current Tetris Guidelines, solo modes are really more of an afterthought. The 7 bag randomizer and preview windows actually make infinite Marathon possible, so the focus is more on quicker time attacks and versus.