So who cares if a non-mining node rejects a block?
Because no other node (including other miners) on the network will then accept any other blocks built upon that bad block and consequently no other transactions built on that block will be accepted. So no miner will be able to spend the coinbase transactions - which means no reward for the miner. Which means they aren't gaining anything, but are losing the entire block reward spending their electricity for nothing. And wasting the capital invested in mining hardware.
? Because the only vote you have is whether to mine on a block or not. This act of mining on a block signifies you accept it. So what do non-mining nodes have to do with this process?
Definitely not. All nodes matter and, as above, they arent votes. All nodes enforce the protocol. Any node that enforces different rules is a fork of bitcoin.
Just put it to the reality test (as Carlton suggested): contact the mining pools and farms and see if theyll use this idea. The answer has been explained above is that they wont, but it seems like you dont understand how bitcoin works well enough to see the problems with the suggestion and so for you, perhaps asking the current miners why they havent already been doing this will help you to understand.
This summer's BIP 148 was a great example of it.