It isn't really a new idea. They have been implemented in CryptoNote (later became Monero) for quite some time.
There is some additional complexity around delivering the necessary out-of-band information (the signatures/scripts themselves) and while it does make the data more "prunable", so to speak, it doesn't actually change the amount of data that needs to be relayed. In fact, it will likely increase it.
This is because a block can be up to 1MB without the scripts, and they are still needed in order to verify the witness, the effective block size with the scripts will increase. Since signatures are 71-73 bytes and public keys are 33 or 65, that basically means in the typical case you offload ~105 bytes per input. So imagine if you have a block with 5000 inputs, that's roughly 525k worth of offloaded data. Now imagine if you fill up the entire 1MB block. I haven't done the math, but my guess is it would be roughly 3-4MB of offloaded data that is needed in addition to the 1MB block.
It's interesting that one of the huge debate points about increasing the block size is all about relay and, as nice as segregated witnesses are for many of the other properties they bring, they do nothing to address the amount of data that needs to be relayed.