What should ATC have done differently?
People often have an idea that ATC actually controls what happens. They just give advisory guidance to pilots, who ultimately decide what to do. A clearance to land or the lack of one does not absolve the pilots from making their own judgments and decisions about how to conduct the navigation of the aircraft, and where and when to land.
Usually, it’s a bad idea to not follow ATC guidance, but in the case of emergencies especially, pilots call the shots.
But the tower at an airport, their job is to supply information, and once you call mayday you’re in total control - you can ask them to do whatever you think will help you save the flight, including (and usually automatically done) diverting all other flights, clearing all runways, and cancelling any departures pending until the issue is resolved.
Pilots have to be trained to ignore ATC as necessary, because planes have crashed because of trying to be polite to requests (and not declaring mayday or pan pan).