If all it does is prevent length extension attacks, then there are much simpler and less risky ways to do that (i.e., MD variants).

Also, your explanation of the sponge structure omits the real difference between it and MD: It is a transform that turns a non-compressing function (f in that diagram) into a compressing function. MD, on the other hand, starts with a fixed-input-size compressing function and extends its domain.

By the way, what do you mean by "Furthermore, Keccak's Sponge design derives security by only allowing inputs to directly influence a subset of the internal state bits."? That's as true for an MD-type construction as it is for a sponge construction. In fact, it's a crucial fact that allows us to build a reduction from, say, the collision-resistance of MD[f] to the collision-resistance of f.