Because with this approach, the entity making the statement doesn't have to cooperate. Even if they want to cooperate, there's a lot of chicken-and-egg problems

Because with this approach, the entity making the statement doesn't have to cooperate. Even if they want to cooperate, there's a lot of chicken-and-egg problems in trust infrastructure that you sidestep by leveraging the existing Web TLS PKI. It would quite powerfully allow you to notarize and audit any statement by any domain. Unfortunately (some would say fortunately) this isn't cleanly possible, so you still have to rely on witnesses and/or magic ledgers in the sky for trust.