Bitcoin doesn't "overhaul" anything and it doesn't have to. It is the
alternative to the existing centralized payment system and offers a brand new definition of money.
Blockchain technology actually has middlemen and the middlemen are the crypto miners. They verify the transactions and get transaction fees plus mining reward in return.
Middleman is usually a centralized authority that you rely on to be able to perform a task. Miners are not centralized, you can become a miner yourself too. But you can never become the bank!
Can you buy Bitcoin(or altcoins) without a bank account? It's theoretically possible, but it's very difficult.
You don't have to buy bitcoin at all. Do you buy your fiat or get paid in it?