The goal is to deal your opponent damage. While you take damage from all of your spells, almost all of theirs will help you reach your goal more quickly. They absolutely have to remove the Eidolon before they do anything else or risk losing themselves the game.

As Burn, your life total matters in exactly one matchup (the mirror).

Have you considered adding Atarka's Command? At it's worst it is Skullcrack and at it's best, it is 6+ damage for 2 mana.

Command is amazing.

I definitely like Command, however I've done 3 color burn before and the manabase was just rough

And revelry is needed in the side.

It's in the side. I meant I don't want to have a 3-color mainboard