In Tennessee in the early 80s there was a “bank” that people put their money into and it went bust. Most people had no idea it was not really a bank and therefore had no FDIC insurance so the people lost their money. They marketed as if they were a bank. A lot of people went to jail but I don’t think anyone got their money back.
A bank needs a federal or state charter and those might require them to use the FDIC. If you don’t use the word bank then it could be that anything goes.
What exactly do you think the Federal Reserve and FDIC do? Most banks operate in the manner you describe. The Federal Reserve and FDIC are safety nets, which step in only in emergencies (though they also act as regulators and mediators).
This is basically what money market funds are. They accept investor money and invest in short-term, high rated debt. They are not insured by the government, although they were briefly insured by the US Treasury as an emergency measure in 2008-2009. Many brokerages offer check-writing services and such against these funds. The returns on assets are typically fairly low, both for the investor and the mutual fund companies.
Well, not really, at least not if that is meant as a reply to the OP’s last sentence “So are there any banks that operate with just their own money?”. In a modern bank, the bank’s own money, i.e. its equity, is negligibly small compared to the bank’s overall operations. Take, for instance, the most recent (2009) annual balance sheet of UBS (5 MB PDF). Total assets of 1,341 billion francs, but only 41 billion francs in shareholder equity - the rest is liabilities. Banks operate with other people’s money (and moneys creation against other people’s money as reserves), not with their own.