While it appears the controller gave clearance, and did mess up, it wasn't the primary reason for the accident.
The runway status lights (there are various kinds) were activated and the driver of the emergency vehicle proceeded anyway. In one of the videos, the "company of vehicles" stopped, the lead vehicle did not.
RWSL is currently installed at Seattle-Tacoma, San Francisco, Los Angeles, McCarran, Phoenix Sky Harbor, George Bush Intercontinental, Minneapolis-St Paul, Chicago O'Hare, Detroit Metropolitan, LaGuardia, Newark, John F. Kennedy, Baltimore/Washington Thurgood Marshall Airport, Washington Dulles, Charlotte Douglas, Orlando and Fort Lauderdale-Hollywood.
skybrary.aero
To cross the runway, the vehicle needed both the REL and Clearance from a controller.