I shoot a lot of flash for my small bird photography.
It is very problematic when out in the field because these are one time opportunities, especially with migratory species.
At a feeder, though, for the most part the same birds are hanging around for at least days, if not a season. And in that case my experience is that they can be "trained" to accept flash. And it does not take long, often maybe a half dozen times.
I'm not saying there is any hard and fast rule here, only that persistence seems to pay off in most cases, especially at feeders.