It does do (a variant of) MCTS. Check it out for yourself. The paper is here:
Appendix B, page 12:
"We now describe the search algorithm used by MuZero. Our approach is based upon Monte-Carlo tree search with upper confidence bounds, an approach to planning that converges asymptotically to the optimal policy in single agent domains and to the minimax value function in zero sum games ."
You are aware that MuZero has tree search hardcoded into it, yes? How does that contradict claim 1?