The new chat history dialog has two issues for me (web on Mac).
1. Once the chat dialog is opened and closed there doesn't seem to be a way to open it a second time. Once the dialog is closed it also claims there are no messages even when there are.
2. Open a game (e.g.
https://www.warzone.com/MultiPlayer?GameID=32531736), click "chat", "public", "load older messages". The dialog shows a mix of team and public messages with no apparent way to tell which are public and which are team.