"Blazer, the Eater of Dimensions'" Continuous ability is a replacement effect. Cards that trigger "when ~ is put into a graveyard" will not trigger when they die if Blazer is on the field, because they never hit the graveyard.
If "Blazer, the Eater of Dimensions" would go to a graveyard from the field, its ability will see itself changing zones and remove it from the game instead. If it would go to the graveyard from anywhere else, including being cancelled, discarded, or milled, it will still be placed in its owner's graveyard.