Imagine a boss that hits 300 times a second for 1 point of damage. In that case, the avoidance tank with 50% dodge probably will avoid around half the hits per second, while the mitigation tank with 50% damage reduction will absorb half of every hit. It's possible the avoidance tank will miss his dodge roll 300 times in that second, but it's statistically unlikely. It's more likely that both tanks take the same amount of damage.
Now imagine the very slow boss who swings once every minute for 25K damage. The avoidance tank will die if he ever fails to dodge twice in a row, but you are almost certainly going to heal him back up in between swings so that doesn't happen. In this other ridiculous extreme, the difference between mitigation and avoidance is also not a big deal.
The streaky nature of avoidance is only a problem when you fail to avoid several times in a row on mobs that hit faster than you can heal, or when the hits can be really large. Without crushing blows, the hits will be smaller.
The other problem with avoidance is when healers just can't get into the rythym of healing and overheal a lot, run OOM, or get caught by surprise. But that varies a lot depending on the skill of the healer and the types of spells.
