> Banks give access to money that you don't currently have with understanding that you will pay them back a premium over time for that service.
By using money that they don't have either. I think that part is important to remember. The difference between a bank and a private person is not that the bank has money to lend. It's that they are allowed to lend money they don't have. If you do that it's called fraud.
You most certainly can lend money you don't have. Of course, you're going to have to come up with it somehow (so will the bank), likely by borrowing it. Which is fundamentally no different than what a bank does, albeit their balance sheets are larger and counterparties different. It's "borrowing" all the way down,and how the financial system works.
>banks can lend money that didn't exist until they lend it to you and that stops existing the moment you pay it back
Of course. How else should the money supply increase but to create it out of thin air? All of it is. Imagine not being able to buy a house until Fred paid mortgage his off? This is a reason why getting off the gold standard was a good idea. What's the point of having a dollar be equivalent to the price of an ounce of gold (which itself is an arbitrary peg), along with all the mechanisms required to maintain that peg? If you want something equal to the value of an ounce of gold, buy an ounce of gold.
This is what Bitcoin proponents get wrong about its use as a currency; a static supply is a problem. Why would you want to limit people's ability to perform transactions in the unit of account? If you can't increase the supply as needed, then you have to find ways to further divide what's out there (which is just the other side of the same coin).
Fractional reserve is what makes possible for banks to lend (part of) the money that clients deposit. In a full reserve system every single dollar in client deposits would be kept in the vaults and the banks could lend only their own capital.
By using money that they don't have either. I think that part is important to remember. The difference between a bank and a private person is not that the bank has money to lend. It's that they are allowed to lend money they don't have. If you do that it's called fraud.