The reason you use Garuda mainly, is because Ifrit spends time travelling to the target. Garuda has no problem just attacking.

As for Contagion, people really don't seem to think about how strong it is with buffs.

Example.

You've just put your DoTs on the target. A few seconds later, Raging Strikes come off CD and Contagion comes off CD. Now, you have two scenarios here depending on your Egi.

Ifrit: Wait until your DoTs are just about to fall off to before popping Raging Strikes to re-DoT, which is a net loss since you've held a CD you should have used instantly as it came off CD.

Garuda: Pop Raging Strikes and at the end of the buff timer when your DoTs are just about to drop, reDoT and use Contagion to get a huge DPS boost.

The second scenario would yield a much higher DPS gain than you would with Ifrit.