You guys don't give the right amount of credit to the writers. Either you give them too much and say that we're supposed to get that Tenzin dies when no signs in the history of television really give you any indication that he activated any of the flags there, or some other shit.
Remember Jet? That was the proper way to kill a character off screen, similar to the Earth Queen. You seem them getting hurt beyond an injury or about to be on screen, then hint at his death to the point the writers are thrusting the character's death unto your face.
If I recall correctly Jet was shown lying on the floor, Katara tried to heal him, somebody said "he is way beyond that", Katara and the rest of the people crying, and a plethora of other hints that Jet was going to die any second now and was beyond coming back. There was little argument that he was unless the writers decided to miraculously bring him back later. Which near the end of the series they said "so Jet did die then" almost as a way to make fun of the crazy conspiracy folks.
The Earth Queen died by showing us Zaheer mercilessly taking out the breath from her lungs, and we see her helpless gasps of air that same episode as she falls on the ground. Then that same episode Zaheer goes and tells people via radio "the earth queen is dead". They made it very clear prior to this.
Tenzin on the other hand was beaten up mercilessly. They were using rocks, splashes of water, and fists/air gusts. That is not the merciless team of Zaheer about to kill someone nor the writers giving enough thrusting into your face hint. You don't kill a character off screen before an episode ends and not have a dramatic moment where somebody sees him dies and cries his name, or somebody close to him starts crying realizing he is dead, or opens their eyes and looks towards the temple with a shocked face before the credits roll. This is not how television works people ffs.
edit: and no saying "i'll fight you while I still draw breath" is not enough. Plenty of people would say the same in that situation, Tenzin is basically making his resolve clear over how much he cares for Korra and his people.