So, about the avatars:
They can only be effectively removed if they're
- Removed from game
- Lose all text
- Some wacky other mechanics
Let's look for each of those how they can be removed.
Removed from game:
- Verore Kidnapper (rather expensive, kidnapper can be removed)
- Kidnapped by Demons (may return after a while)
- Agent Coyle, Ascended (needs to battle, coyle will likely die as well)
- Heavenly Wish (rather expensive, only in 3p OoS)
- Ethereal Fusion (who the puffy plays Ethereal Fusion)
- The Calamity (very expensive)
- Gorgon Beam (rather expensive, only hits the battlefield)
- Xi, martyr (very expensive, is DoD)
- Spiritualize (only hits the battlefield, is DoD)
- Demonize (only hits the battlefield)
- Descension (I ain't even mad)
Lose all text:
- Defiance (your character becomes stronger)
- Humble (is from DoD)
Some wacky other mechanics:
- Altar of binding (hard to hit right, altar can be removed)
- Struggle for power (not easy to pull off)
- Suza, Angelic Siren (can be removed, only hits the battlefield)
Aside from that, FD has some nasty red tape, humble and exhaust. But they don't completely remove the looming thread of an avatar.
From the above mentioned removals, most are expensive removals, the ones that aren't have limitations you can play around. The (arguably) most threatening card is Humble, but that's a DoD card itself. It's true that (almost) every faction has an answer to the avatars, but the game would no longer be challenging if they don't.
I agree that triple DoD has some problems, but that's the case for almost 3p deck. I don't believe the avatars or the removal are the problem here.
It's not that easy to completely remove the avatars. Triple factions are rather weak anyway.