If you are a guy in your basement, and that you want to find a solution to keep the chain coherent, and you choice between à simple well proven determinstic solution , or a solution that is completely off chart, super costly, and risky, why he would choose the second ? Why going through all this bother with pow and block reward who introduce huge complexity ? Why ?
What deterministic solution ? There isn't any that isn't centralized or permitted. Proof of Stake was a possibility, but Satoshi was facing the problem that he was the only stake holder in the beginning. He would have had to sign all blocks by himself, and unless someone actually GOT COINS FROM HIM, there was no way to get a second stake holder.
That could be just be as simple as selecting block and tx based on which have the lowest hash. Period. No pow, no reward, no mining craze.
The problem is, WHEN do you consider that transaction A is the valid one ? How LATE can transaction B be propagated and WIN from transaction A ?
Suppose I pay you 100 BTC. You observe transaction A on the network paying you. How long do you wait before you consider that this payment is secure ? Suppose I buy a car with that. How long do you wait until you let me have the car ?
Suppose that the next day, I make a new double spend payment to myself. I can modify my receiver addresses until I find a payment that has a smaller hash than transaction A. I call that transaction: B. I now transmit B on the network. As B has a smaller hash than A, the consensus tells us one should take B over A, and finally, your transaction is eliminated.
Ok, but one day later, we don't accept this any more. Ok, but how long do we have to wait ? At what point do you consider that A is definitively the accepted transaction ? After 30 minutes ? But what if B comes in after 29 minutes for Joe and after 31 minutes for Jack ? Joe and Jack will now disagree FOREVER over what was the right transaction ? If you connect to Joe, you see your transaction reversed, while if you connect to Jack, you see your transaction not reversed ?
--> this is the consensus problem. It is already difficult if most players want to play honestly. It becomes very hard if you get a sybil possibility of 90% of the nodes conspiring to game the system (90% of nodes in the hands of one entity).
Suppose that I transmit transaction B almost immediately after transaction A, but I fire up 90% of nodes that "ignore" transaction B. You will probably not see transaction B, and you think that after half an hour, you are safe. Then I switch off my sybil nodes. The rest of the network has preferred transaction B. When you try to spend your coins a few months later, your right to spend doesn't exist on most nodes, because they had rejected A, and chosen B, and forgot about A. You are the only one remembering A, thinking it was right.
Satoshi found a kind of solution with PoW. It is a clunky solution, but he needed one.