Aside from MCTS being a different tree search method, there is no 'closing of the loop'. In regular MCTS, it is far from unheard of to do the random playouts with instead some 'heavier' heuristic to make the playouts a little better estimators of the node value, but the heavy playouts do not do any kind of learning, the heuristic you start with is what you end with; what makes this analogous to policy iteration (hence the names for the Zero algorithm of 'tree iteration' or 'expert iteration') is that the refined estimates from the multiple heavy playouts are then used to improve the heavy playout heuristic (ie. a NN which can be optimized via backpropagation in a supervised learning of board position -> value). Then in a self-play setting, the MCTS continually refines its heavy heuristic (the NN) until it's so good that the NN+MCTS is superhuman. Then at play time you can drop the MCTS entirely and just use the heavy heuristic to do a very simple tree search to choose a move (which I think might actually be a minimax with a fixed depth but I forget).

The branching factor for games like Go is too high. Alpha Go Zero uses a stochastic policy to focus its attention on useful branches. This policy is optimized alongside the predicted value for each node.

Minimax algorithm with alpha-beta pruning is an exhaustive search algorithm. MCTS is a probabilistic algorithm that optimises for converging faster for a partial "good enough" result.

Introducing MCTS for Go playing programs in 2006 brought upon a revolution in playing strength. For the first time Go programs stood a chance against a dan-level amateur.