Does Monte Carlo tree search qualify as machine learning? To the best of my understanding, the Monte Carlo tree search (MCTS) algorithm is an alternative to minimax for searching a tree of nodes. It works by choosing a move (generally, the one with the highest chance of being the best), and then performing a random playout on the move to see what the result is. This process continues for the amount of time allotted.

This doesn't sound like machine learning, but rather a way to traverse a tree. However, I've heard that AlphaZero uses MCTS, so I'm confused. If AlphaZero uses MCTS, then why does AlphaZero learn? Or did AlphaZero do some kind of machine learning before it played any matches, and then use the intuition it gained from machine learning to know which moves to spend more time playing out with MCTS?
Monte Carlo Tree Search is not usually thought of as a machine learning technique, but as a search technique. There are parallels (MCTS does try to learn general patterns from data, in a sense, but the patterns are not very general), but really MCTS is not a suitable algorithm for most learning problems.

AlphaZero was a combination of several algorithms. One was MCTS, but MCTS needs a function to tell it how good different states of the game might be (or else, it needs to simulate entire games). One way to handle this function in a game like chess or Go is to approximate it by training a neural network, which is what the Deep Mind researchers did. This is the learning component of AlphaZero.