My main reason for thinking this is that the idea of simply having transactions validated across the blockchain (feel free to fry me if I am being ignorant about how things work) seems to be enough to make the entire concept work. Mining, the long hashes, all the other stuff I barely even fathom, it all seems like just painting flames on your bike to make it go faster. It seems unnecesary. If I make a transaction and that transaction has to be confirmed by dozens of nodes that track all the transactions, isn't that a pretty hard thing to outsmart already?
I'm not saying the other things should not be done, I just would like to understand the value they add to the whole process. Because every wallet essentially meeting up to check if a transaction is a-okay seems like the best system to me, all on its own!
Nope. Proof Of Work is essential in making Bitcoin work in the first place. It acts as a consensus mechanism among nodes to decide which are the transactions to be accepting and valid. It builds the blockchain and provides security to the transactions that are included in the blocks; as the confirmation increases, so does the security as it is the amount of work that an adversary has to do in order to reverse those blocks.
Without a system like that, someone can just broadcast two different transactions spending the same coins. Which transaction should the node then consider to be valid? How does every node on the network consider that transaction valid and not the other transaction? In Bitcoin, transactions are thus included in blocks and the nodes will always follow the blockchain with the largest cumulative proof of work. This ensures that the same coins are never spent twice and all of the nodes will recognize the same set of transactions as valid.