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The Phoenix Gem relic states the following:

The first time the bearer is slain, roll a die for each unit (friend of foe) within 3" of this model... If at least one mortal wound is inflicted as a result of this, then this model is not slain, but remains in play with a single wound remaining...

This is interesting, because it triggers when a model is slain, but specifically states "is not slain". I have to think that you don't continue to take any remaining wounds, as this would trigger after the shooting/fight/power is resolved but I'm not sure if it counts for abilities that trigger when a unit is slain. What are your thoughts?

Just more fuddled stuff because Ynnari is dumb and poorly thought out.

Since the unit is not slain, Soulburst would not trigger.
Phoenix Gem triggers when the model is slain. Since the model is still on the table for phoenix Gem resolution, the unit is still there. Once the model is removed, then the unit is slain.

The Phoenix Gem triggers "when the model is Slain". Strength from Death triggers "Each time a unit is completely destroyed".

These are not the same thing. "Each time a unit is completely destroyed" means the last model has to be actually removed as a casualty, not just be "slain".

Being slain is normally synonymous with being removed as a casualty, but they are not the same, since effects like the Phoenix Gem or that Tyranid one exist.