Some definitions to be used in the following:
Mitigation: Guaranteed reduction to incoming damage. In FF14, this is Def/MDef/Resists and shield oath. Cooldowns can be included as temporary mitigation, but in gearing we're generally concerned with static mitigation.
Avoidance: Non guaranteed reduction to incoming damage. Block, dodge, parry all fit this bill and by extension strength and dexterity.
EHp: Effective HP, this is your WORST case scenario hp. If you had a die that always rolled 0's, this is your HP. It's equivalent to your HP * (1/(1-Mitigation(as decimal))).
AHp: EHP * (1/(1-AverageAvoidanceReduction(as decimal))). This is your average HP total. This number is bollocks and generally useless as a metric.
Alright so to start off, as a general rule of thumb:
Mitigation > Hp > Avoidance - You've heard this before, and it's true. 95% of the people wanting to tank in any game should just look at this, accept it, and move on.
There are two ways a tank can die, from burst damage or from attrition. Attrition is what takes place when average incoming damage is greater than the average incoming healing. This is generally an indicator of a severely undergeared tank, severely undergeared healers, or a mismatch in party set up and/or mechanics(not enough healers, 1 tank tanking a two tank encounter, not responding to a mechanic.).
Mitigation and Avoidance survive Attrition, HP does not factor in(well, beyond the innate regen boost you gain from hp pools)
Burst damage is exactly what it sounds like, it's a burst of high damage attacks that when combined together one shot or nearly one shot a tank before the healers can react. It's characterized from a tank going from 100% to dead in less than 2 or 3 GCDs. There is nothing that can be done from either the tank or healers perspective beyond the possibility of blowing a cooldown. When preparing for burst damage, you have to prepare for the worst case scenario. The worst case scenario is, by definition, when you did NOT avoid any of the damage in the burst damage combo.
Mitigation and HP(see Ehp) survive Burst, Avoidance does not factor in
What Avoidance DOES do in relation to burst damage is reduces the chances that it will happen, which in turn gives your raid less chances to fail. However if your EHP is not sufficient to survive the Burst damage, then it doesn't matter if you reduce the chances of it happening by half. When it happens, your face is going to plant into the ground and your raid is going to wipe.
TLDR:
The only thing that kills tanks in a raid group that is properly formed and properly performing mechanics is burst damage. Burst damage must be calculated at it's worst case scenario, which by definition denies any avoidance. Therefore if you're dieing to burst damage (such as Mountain Buster + 2 AAs) the only method to counter it to reach a 100% chance of survival is more mitigation or more HP. Avoidance will make the truck hits less likely, but only EHP will ensure you survive them when they happen.