The part that was left out is the fact that conjugating is the group action of a group on itself.

in his example, let's say there's some 3-cycle (123) the cuber knows. But really, the cuber wants to transform like (479).

Conjugation being equivalent to a group action implies the cuber only needs to find a group that permutes 1->4, 2->7, 3->9.

Let's say one is found: (14)(27)(39)...

gTg' => [(14)(27)(39)...](123)[(14)(27)(39)...]' = (479)