The phrase "remove their own trigger" is misleading. Reactions and other such effects "interrupt" their triggers. You can cast shield and still get hit, for instance.
By this logic, you can get reduced to 0 hit points in wildshape and the moment when you revert to your own hit point total can be seen as an interruption of the normal steps invoked upon reaching 0 hit points. In fact, that's almost exactly what it is.
Same with disintegrate. Disintegrate is the trigger, which the wildshape reversion interrupts, thereby denying it the "at 0 hit points" trigger unless the remaining damage, per the wildshape rules, is enough to reduce the druid to 0 hit points in their true form.
Basically, disintegrate can't invoke it's "reduce to ash" trick until a) the target is reduced to 0 hit points from the damage dealt and b), the spell is done dealing damage. Effects that interrupt the damage of the disintegrate spell don't trigger the "reduce to ash" trick.