Cosmo
Now I see what was on your mind.
The issue seems to be the presumption of "flame-out".
Let's be careful about this. A true flame-out implies the burner going unlit and the engines spooling down to windmill. It is extremely rare except in cases of fuel starvation. Yes if so it does require a time consuming restart procedure.
Birds are not likely to cause such a flame out. In most cases it is the pilots decision to turn off the flame to secure a misbehaving engine.
While I'm not a 100% sure yet in the Hudson accident it doesn't seem to me that they ever shut down the engines.