The Importance of Profit Motivated Banks

Banks are almost providing a public service, you know, safe place hypothetically to keep you money and also lend that money. Do you think there's an argument or even perhaps it would be easier to introduce a bank that's just run by the government? In Europe, we still have a lot of banks that are public or semi-public. So I'm not sure it would be such a good idea to have a lots of public banks.