Could Bitcoin be based on accounts instead, more like banks?
It could, yes. Ethereum works on something similar to an account system, at least as far as ethereum balance is concerned. Both models are valid, and have their benefits and drawbacks.

The UTXO model allows for great parallelism and privacy. Each transaction can create UTXOs to new addresses, and different UTXOs can be spent independently and parallely. 

The account model requires some form of ordered execution (achieved via a nonce in ethereum). Moreover, a single transaction cannot combine balance for two different accounts in current implementations, while you can combine UTXOs from as many addresses are you wish in a single transaction.

As to why Bitcoin chose a UTXO model over a balance one, I don't think an authoritative answer exists on that. It's what the creator(s) of Bitcoin started with, and there is no pressing reason to change it.