Fawkes was genuinely loyal to Dumbledore and even protected him once by taking Avada Kedavra on itself (and then being reborn from its ashes).
Why didn't Fawkes protect Dumbledore once again when Snape
cast Avada Kedavra on him
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As Harry will learn almost a year later when watching Snape's memories in the Pensieve (and we as the readers along with him), Fawkes was present during the discussion of how all of this was supposed to play out:
Snape raised his eyebrows and his tone was sardonic as he asked, “Are you intending to let [Draco] kill you?”
“Certainly not. You must kill me.”
There was a long silence, broken only by an odd clicking noise. Fawkes the phoenix was gnawing a bit of cuttlebone.
“Would you like me to do it now?” asked Snape, his voice heavy with irony. “Or would you like a few moments to compose an epitaph?”
“Oh, not quite yet,” said Dumbledore, smiling. “I daresay the moment will present itself in due course. Given what has happened tonight,” he indicated his withered hand, “we can be sure that it will happen within a year.”
Deathly Hallows, Ch. 33
As you rightly said, Fawkes is loyal to Dumbledore, that means he follows his wishes. Dumbledore is already dying (apparently healing tears are not enough to counter the curse Voldemort put on the Ring), he wants Snape to do it quick and painless and they are waiting for a right opportunity. Basically they are waiting for this exact situation. Fawkes knows all this and therefore does not intervene.