Earlier quoted context omitted.
Almost like...their reflex is to fly until they are out of the smoke. Which is adaptive when dealing with small fires, and lethal when they are unable to fly far enough to get out of the smoke.
I'd be happy to read any reputable paper, article, or discussion thread you can find about it. Seems like something that would be falsifiable and has been researched already, so not worth debating about in the hypothetical.
Multiple passages in it are relevant to this issue. Birds do try to fly away from fire. We have evidence that smoke inhalation is worse for birds than mammals (which would mean that birds would need to flee smoke, and not just the flames). And the possibility of birds dying of exhaustion from trying to fly away was already a concern to wildlife conservationists.
None of which is proof. But it is all suggestive that I'm making a reasonable guess as to why they are dying.