I mean, seriously how can you bring down Isengard and not show the demise of Saruman at the end of The Two Towers or the beginning of King in its threatical form? And then when you put it at the beginning of the King Extended, it's logically and tonally a mess and feels really tacted on.
The Fellowship is just pure Tolkien and even in the scenes they changed slightly, it's still more Tolkien than the rest of the trilogy.
Just give me The Fellowship Extended, Revenge of the Sith, and A New Hope if I had to be strained on a island.