First of all the, media always gets these stories wrong. Second generators make electrical power for the aircraft systems. It isn't for the engines. The generators are run by the engines. Now if I was flying and something goes wrong and I'm thinking of diverting I would probably slow down, go through the checklist, talk to dispatch, and decide on a plan.
The idea that the NYT is using flightaware to report this is hilarious. Surfing the web is a substitute for journalism now.