Let's look at Phoenix Fire transport for a minute, shall we?

First off, Dumbledore uses it. That needs no documentation, I hope.

Secondly, he uses it to transport Trelawney:

“He is coming,” said a huge hollow voice that cut through all conversation like a sword of ice. “The one who will tear apart the very—” Dumbledore had leapt out of his throne and run straight over the Head Table and seized hold of the woman speaking those awful words, Fawkes had appeared in a flash, and all three of them vanished in a crack of fire.

Thirdly, he uses it to transport Hermione and Harry (Demented & post-trial):

Harry caught fire and went out and blazed up somewhere else; and just like that he, and the Headmaster, and the unconscious form of Hermione Granger held in the Headmaster’s arms, were occupying another place; with Fawkes above them all. ... From a distant corner there was a flash of green, and from out of a fireplace strode Professor McGonagall, brushing herself off slightly from the Floo ashes.

But not McGonagall. So ve can transport Harry, Hermione, and Trelawney, in addition to Dumbledore, but not McGonagall? Why?

I searched for "crack of fire" and did not see any other examples of people traveling by phoenix fire, nor did I remember any. If you find any, I would appreciate a quote.

I am going to axiomatically assume that phoenixes cannot exclude their master from phoenix fire until they decide to leave. If you have any counterevidence, then I would appreciate hearing it.

“The phoenix chooses but once,” said the old wizard. “They might perhaps leave a master who chooses evil over good; they will not leave a master forced to choose between one good and another.

So they might leave their master, but until then, we exclude Dumbledore from any criteria that may exist about who phoenixes may transport.

It seems to me, as figures of Good, perhaps phoenixes can only transport the innocent, which would explain Hermione and Harry, for the most part, but not really Trelawney. It's possible instead that they just won't take Bad, or even Evil, but that hardly applies to McGonagall.

So in what definition of Good or Innocent would apply to Trelawney but not McGonagall?

Is it, perhaps, killing?

We know that neither Hermione nor Harry have taken a life, especially if MOR!Voldemort didn't die the way of canon. Trelawney frankly doesn't seem competent enough to get out of a serious fight alive, so I would assume that she hasn't killed. McGonagall, on the other hand, we know fought Voldemort, because she remembers crossing wands with him in battle.

She had encountered the Dark Lord four times and survived each one, three times with Albus to shield her and once with Moody at her side.

This implies that she fought at least four times, likely many more, and it seems unlikely in the extreme that she managed to avoid killing people every single time.

So my conclusion is that phoenixes cannot transport people-not-their-masters who have lost their Innocence by killing. It is possible that instead the barrier to McGonagall joining them is not about any facet of her specifically but rather that Fawkes can only transport three people at a time. Either way, this will become important in time, due to Conservation of Detail.

Thoughts?