This story doesn't draw a point between Emet and the bird.
It clearly does. Hythlodaeus literally says "it's as if you wore the feathers yourself," that is as explicit as it's possible to describe a parallel. The phoenix in the story houses a soul filled with regrets that rages against it's death in a self-destructive struggle. This is a direct parallel to Emet and the Ascians after the Sundering, as they're destructively struggling for the rebirth of a dead world and it's people. To The Edge even calls back to this specifically with many of it's lyrics. And while we're meant to view this as tragic and regrettable, and I even think WoL is supposed to parallel Emet in this story as it's his task to end the phoenix' suffering, the intention was to paint them as pitiful yes, but not terrible. Of note is the parallel Emet himself draws, in that he describes mortals in terms akin to the phoenix, as thrashing about in their broken lives and repeating their mistakes over and over, and so we see that from his perspective his task with them is the same as with the phoenix, and yet from a greater context we also see that Emet is no different.

There are a ton of connections being drawn in the story, and I think it's too reductive just to say "he had empathy for the bird but disgust for Varis, so the point of the story is to show how he changed into a bad guy."