No, actually, kakashi didn't die from Shurado and Tendo, yes he suffered to much damage from them but he died from losing to much blood i guess, depleted chakra and stamina .. If katsuyu came and aid him fast, he will still stay alive .. but yeah, technically, he was killed by the pains .. (Remember, he used kamui to evade the nail that is intended to pierce his forehead and used another kamui to transport Shurado's missile to keep choji alive and that makes him deplete his remaining stamina and chakra and died) ..