We can't make non-safety related PAs while the airplane is moving on the ground, or in non-cruise flight below 10,000' It violates the regulation on sterile cockpit. 14 CFR § 121.542

It doesn't sound like this UX crew was very good with passenger announcements overall, but we don't know how busy they were, when not in sterile cockpit, between making the decision to divert and landing.

The way you work around 121.542 during ground operations is that you wait until the airplane comes to a stop and you make the announcement them. It's also a good idea to wait until after any runway crossings have been completed so that both crewmembers can hear any ATC instruction regarding crossing, or holding short of, any runway.

The only announcement I make during taxi is the required "Flight attendants, please be seated for departure". For anything else, such as a tow-in, I wait until the airplane comes to a stop.