Even if Matthews doesn't sign an extension right away on July 1, 2023, it doesn't mean he's wants to leave as a UFA or that he needs to be traded. In the most recent offseason Colorado didn't re-sign Nathan MacKinnon right away on July 13, 2022, which was the first day possible for him to have done that. He signed his extension on September 20, 2022.
So, there is a chance Matthews could sign a contract extension with Toronto after July 1, 2023, and no one should freakout if it doesn't happen on that specific day.