^Except, again, the first singularity Nero fell into threw him back into the past, where he was able to wreak havoc with his anachronistic technology. And this
singularity was generated by the exact same technology. So how is it logical to assume this singularity would've killed him, rather than sending even further into the past where Earth and the other Federation planets would be defenseless against him? Kirk had to make sure that threat was neutralized. He couldn't take the risk that Nero would be sent further back in time, and had to prevent that by any means possible. He would've preferred to do it by rescuing Nero, but Nero took that option off the table. So really, he had only one option left when protecting the whole timeline was at stake.
(And yes, I know the difference is that this time the singularity opened up directly around the ship, but it was a damn big ship, and enough of it could've survived intact to allow Nero to survive the trip and salvage the weapons he needed. After all, he was imprisoned in the 23rd century for 25 years while the Narada
repaired itself. Time travelers tend to have plenty of time.)