in order for a transaction to be added to the pool of valid transactions that will be added to the chain block by the miners, at least 51% of the nodes/peers need to review that transaction and see that the public key issuing it A) has the funds to issue it, and B) has not spent them in another transaction.

This is incorrect, there is no requirement for a certain number or percentage of network peers to review or validate a transaction before it is included in a block. A transaction is either valid, or not.

Validation is done at the scale of each individual node: each node can validate a transaction on its own, and know that it is valid, without relying on any other node to confirm this info for them. An overview of the transaction validity rules can be found here. Only valid transactions can be included in blocks, so miners will validate transactions before including them in a block they are mining. Including a non-valid transaction would make the block invalid, so miners are incentivized to make sure they only include valid transactions.

In fact, miners can include transactions that were not broadcast to the network first, so that the rest of the network will first hear about the transaction when they receive that miner's new block (assuming the miner finds a valid block, of course). As long as the block is valid (which requires all transactions to be valid), this is just fine, no 51% majority is needed.