The objection to the Phoenix isn't about the underlying metaphor, but the increasingly lazy justifications for Jean Grey's return in the comics. No one just says "This is the mechanism by which she can die and come back" and leave it. (You also have the deeper problem that it was later changed to the idea that the Phoenix was a separate being that Jean Grey which further complicated later story lines where Jean Grey died and came back without the benefit of the Phoenix Force. But that's another thing entirely.) My joke, as well, was less about the Phoenix itself and more about the revolving door of death Jean Grey has been since the 70s. She once died and came back and died again within the same 5 issue storyline. It's about devaluing death. And I love the Phoenix, I love where they went with the storyline. But lately there hasn't been a "Price" for coming back. People just plop out of the dirt and say "Alright, alright, alright" like so many McConaugheys.
So yes, the idea of using the Phoenix as a metaphor is of course linked to life, death, rebirth. But... the comic character named Phoenix went from interesting concept of said metaphor to big joke for death in literature.
And besides all of this, my joke was really more surface-based about two story lines whose covers/advertisements heavily hint at an impending resurrection of a beloved, red-headed kick ass woman. Though what the end result of both Eternal Tide and Avengers Versus X-Men are both unknown.