I had the same issue just tonight. (All my settings were good.)
I fully closed MCC, reopened it, and everything was fine.
My game chat works until someone sends me a Xbox live party chat request. If I enter party chat, in game chat does not work afterward. It does not matter if I manually switch to game chat while leaving the party chat active, or if I exit party chat. Restarting the game doesn’t work. Afterward, I must perform a complete power cycle of my Series X to have game chat function again.