We need to reevaluate who 'sold us a bill of goods'. You have abdicated your authority in the network. In reality, non-mining nodes are irrelevant. Indeed, in the original wallet, nodes mine - period.

https://github.com/trottier/original-bitcoin/blob/92ee8d9a994391d148733da77e2bbc2f4acc43cd/src/main.h#L795
 
// Nodes collect new transactions into a block, hash them into a hash tree,
// and scan through nonce values to make the block's hash satisfy proof-of-work
// requirements.  When they solve the proof-of-work, they broadcast the block
// to everyone and the block is added to the block chain.  The first transaction
// in the block is a special one that creates a new coin owned by the creator
// of the block.
GitHub
trottier/original-bitcoin
original-bitcoin - This is a historical repository of Satoshi Nakamoto's original bitcoin sourcecode

(emphasis added)