their triggers leave the stack
That's where your reasoning goes wrong. Once an ability is on the stack, it's 'separated' from the original object:
112.7a Once activated or triggered, an ability exists on the stack independently of its source. Destruction or removal of the source after that time won't affect the ability. Note that some abilities cause a source to do something (for example, "Prodigal Pyromancer deals 1 damage to target creature or player") rather than the ability doing anything directly. In these cases, any activated or triggered ability that references information about the source because the effect needs to be divided checks that information when the ability is put onto the stack. Otherwise, it will check that information when it resolves. In both instances, if the source is no longer in the zone it's expected to be in at that time, its last known information is used. The source can still perform the action even though it no longer exists.
Removing the source of the ability does not help countering the ability or removing it from the stack. You'd have to remove the player (see rule 800.4a) for something like this to happen. Of course, you can counter the abilities with cards like Stifle because you do get priority before they resolve.