I'm not saying we stop checking signatures. Only that they not be include in blocks, to save space. Signatures are big and excluding them would save a lot of space. The next block is only valid if it respects original transaction data, which nodes still have in their mempools, so I don't see how miners can steal anything. New nodes, or nodes that have been offline for a while, would only need to trust that the tip of the blockchain is as good without signatures as it is now, which it would be.