I don't think it states that Link is killed, just that he was incapable of stopping Ganon. That may imply death, but it doesn't necessarily mean that is the case.
It actually does, or at least it says he's "defeated." It says it in the HH and the video in the OP says it too. His defeat results in Ganon getting all three pieces of the Triforce, but then Zelda & the sages seal him the dark world, cue LttP.
If he wasn't killed/defeated I don't think Ganon would have had any way to get the triforce of courage.
Kill is just a crude word for Nintendo games.