We have plenty of rockets that can get large payloads into escape velocity. The Atlas V was used to get the 8900 lb Curiosity rover mission on its way to Mars. If it can get a 4 1/2 ton payload on its way to Mars, it can get a smaller payload to an asteroid.
That said, this looks like unscientific scaremongering from a questionable Russian website.
I am.
Remember the space nuke tests of the 50s. And yes, A Sat5 type delivery system would be better to hit it further out. But as you say, none available.
An ICBM will achieve sufficient altitude to destroy the thing before impact and negate the vast majority of damage that an actual impact would cause. Plus they are available, deployable and pretty much ‘ready to go’.
No, not a perfect solution, but lacking that, what else ‘could’ we do in the available timeframe?
“What you need is something like a Saturn V, that could put a large mass onto escape velocity. But we haven’t had one of those since 1973.”
IIRC they don’t even have all of the plans for the Saturn V, and wouldn’t know how to build one even if they wanted to. Sad.