You know, I was looking at this thread and wondered about it and decided that the engine probably did not flame-out from air-starvation. As greasespot put it: jet engines are really good about drawing in air.
I'm not sure what the fuel management system is like on a 737-300, but could fuel starvation have been the problem?
We had a DC8 (same base engine) flame out because the flight crew failed to properly configure the fuel system prior to take-off. Fairly easy to do on an 8, don't know about a -300.
We wound up changing that engine because of the thermal shock of going from TO
to not running without cool down. He was well into the take-off roll when it flamed-out. Was also rather cold out as I remember.