harry potter - Why didn't Fawkes rescue Dumbledore in the Half Blood Prince?

I'm rather surprised, the most obvious answer here seems to be missing.

How?

How could Fawkes have helped him? A phoenix is a powerful magical creature, yes, but it has shown no battle prowess beyond that of a strong bird. Fawkes hurt the basilisk, sure, but only by attacking it's eyes, which doesn't require any great strength. It has no offensive magical abilities that we've seen. A bird versus a bunch of armed Death Eaters - not exactly a fair fight.

It could show up and swallow the Avada Kedavra from Snape, sure (though of course by that point Dumbledore wouldn't want it to) but then what? You'd be back in the same position, save with a baby Fawkes (that the Death Eaters could perhaps kill for real, somehow? Maybe I should ask a question about phoenix mortality...). Dumbledore would still be nearly dead and wandless, surrounded by Death Eaters.

With or without Fawkes, Dumbledores fate was sealed the moment he immobilised Harry.

AMENDMENT

It's been pointed out to me that Fawkes could have teleported Dumbledore away. I suppose this is true, though in canon we have no explanation of how and why Fawkes appears when he does. It's likely he has some form of autonomy, since he appears to Harry in book 2 (though at a stretch maybe you could claim Dumbledore had some sort of "help-student-who-needs-it" spell going on). He's with Dumbledore when he flees in book 5, and then appears during the duel (again, it's not clear if he appeared of his own will or was summoned, though Dumbledores calm demeanour suggests summoned to me).

Not much info to go on! Perhaps it takes energy/strength for Dumbledore to summon him that he simply didn't have in the tower?

Perhaps the potion blocked the connection between the two? That would make sense, since Voldemort didn't want anyone being rescued from the cave.

Unfortunately, without any more canon info it's largely just speculation. Hell, perhaps Fawkes was out on some Order business, or even just sleeping!

One slightly more informed possibility is that Dumbledore judged the risk of leaving an immobilised Harry - even an invisible one - with the Death Eaters too high. After all, if one were to trip over him he'd be dead. Or I could absolutely see a scene where Greyback stops mid-sentence and starts sniffing the air, moving towards Harry...(do werewolves have heightened senses in the HP world?).