Obviously it was virtually impossible to save the crew due to the singularity. But they still should have tried or at least considered the implications of their actions for two reasons, a moral and a political one.
First, the interspecies equivalent of human rights. You do not shoot people "like dogs", neither today nor in a better future (obviously AbramsTrek is reactionary, its future is worse than our present), you do not kill an entire civilian crew just because their leader is a mass murderer. There might very well be people in it who are opposed to Nero's path.
Second, while Kirk knows some tidbits from Spock senior he doesn't really understand the situation and doesn't really know for sure that the ship is from the future so saving the crew is politically prudent. The relations with the Romulans are probably tense anyway.
Instead we got the very opposite, a pubescent Kirk who was basically happy that Nero declined his offer such that he could "pew-pew" him, a character devoid of decency and intelligence. Spock did not even pretend to care and his implicit "come on Jim, kill that sucker who killed my mummy" stands in stark contrast to the Spock from TOS who tells Kirk to kill Mitchell or let Keeler die when it is the only option.