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 because the confirmation process isn't changed.
There is an argument to be made that stripping signatures would make the transactions themselves impossible to verify, and would force software applications to collect entire blocks and verify those.
Storing the signatures ensures that nobody can find a vulnerability to manipulate the monetary supply in a way that other people cannot detect.