I don't think amaterasu burns fast enough to deal with the core and by the time it does Itachi would have been crushed....
I think what he are forgeting is that the paths can't comprehend pain and so Tsukuyomi will not have an effect on them, because it breaks the mind which they lack.
Against Nagato, well Nagato's Genjutsu can overwhelm Itachi Just as Itachi's can overwhelm his.
Tsukuyomni is also not a skill that Itachi can spam whereas Nagato can use his Genjutsu on many bodies.
Next Up, Itachi himself said it would take a powerful attack to destroy that black sphere, the sphere it's self has a lot of pressure on it from the surrounding rocks so you know it will only be destroyed by something on the scale of a bijuu bomb, Naruto could have used a rasenshuriken on it but he did not, he used a bijuu bomb along with Bee, so I doubt Amaterasu flames, a slow burning Katon, would work.
Also the Amaterasu could end up harming Itachi since he is being dragged in the same direction, by the time Amaterasu burns through that sphere Itachi's Susanoo would have been completely destroyed along with him.
Also whats with the start of that vid, the six paths of pain could have worked like this.
Deva pulls everything around him in, while Deva is on standby to suck in any Ninjustu, this would pull Itachi out of Hiding.
Meanwhile Human path sucks his soul out, it's funny how that vid got rid of human path so quickly.
Prince Charles Nagato has extensive experience fighting in a war against Hanzo and his army, we have no reason to say he would fall for a weak Genjutsu like that.
Nagato punches a hole through Susanoo and Itachi from 50 meters away.