I don't exactly understand what's your point here?

Truck 1 and Company were cleared to cross Rwy 4 at D while Jazz 646 was on short final.
Usually only the leader is in contact with the ATC and the rest follow the leader and monitor the radio only, supposedly there were a couple of vehicles crossing at the same time.

Should they have called out "cleared right" before proceeding? yes but I am not driving the fire truck, what do I know? They were responding to a separate emergency, the visibility may not be great at the time or they misjudged the landing light on the Jazz as a departing plane lining up on Rwy 4.
The fault is not to be blamed on a single induvial but a systemic failure.