Shaq is not a vast upgrade over Perkins. Perkins is vastly better defensive player, because he can actually defend the pick and roll, and honestly as ridiculous as it is to say, Perkins has better range.
On top of that, I don't think Shaq converts on a much better clip than Perkins does around the hoop. Shaq is shooting 63% from the field, Perkins shot 60% last year.
Honestly I don't think Shaq plays the offensive side of the ball much better as an individual, and he's not a better defensive player, period. The one big advantage Shaq has on Perkins is that Shaq is massive. He draws way more fouls. I wish we had a "fouls drawn per shot attempt" stat, but since we don't, some basic excel data will show us:
Shaq: Career FTA's/25 mins 6.7, this season 4.2
Perkins: Career 2.1, last season 3.3
Not a huge disparity, but one that is pretty important.