Banks are "Trusts"... No? You trust your funds will be there.
bank
noun
1. an institution for receiving, lending, exchanging, and safeguarding money and, in some cases, issuing notes and transacting other financial business.
2. the office or quarters of such an institution.
3. Games.
a. the stock or fund of pieces from which the players draw.
b. the fund of the manager or the dealer.
4. a special storage place: a blood bank; a sperm bank.
5. a store or reserve.
You are arguing semantics, and failing. A bank is nothing more than a storage location to keep things of value. Next you are going to say that credit unions are not banks... A blood-bank is a bank... Some even pay for blood, exchange blood for money, OMG!
The initial value of BTC came from money spent on assets, and time labored without pay. Debt. The same thing that drives it now. The same thing that gives money its value.
After mining, the miners still have to run, to do the transactions. Us, being the banks, have to depend on them, or buy assets to keep the network moving. The cost is our assets and operation expenses in electricity and time. Our payment is fractions of fees, after mining has stopped.
A bank-note is a check for $1.00 or $5.00 or $10.00 ... ... It is owned by the bank, given/lent to you, and has no actual physical representation as an asset beyond what you "think" it is worth, to you. Banks will not give you anything other than another dollar, for your dollar... Actually less, in most cases.
We stop using dollars, and the thing in your wallet is only of value to you. The bank won't take it, or they will, but you will get nothing for it. They have nothing to give you. They won't even give you the pen at the teller, because they don't want your dollar. It is worthless to them if you don't want it, and if they can't find anyone to use it.
It has happened before, and it will happen again. Banks are an old inefficient dinosaur that can't afford to keep up. It costs them too much of your money to maintain. Currency will be tulip-bulbs in the future. Something like BTC, birthed from BTC, will be the future. (For now, it is the future. FIAT is undoubtedly the past.)
You are right, it is not a "money lender"... That is not the only type of bank. Not all banks lend your money out to strangers or their friends, or themselves. If that is your idea of a bank, then continue to use those places. They are doing you wonders, decreasing the value of your dollar. The one in your wallet. (Wallet is a style name, like purse, or vault, or safe, or pocket. They are all storage containers, and thus, banks. Funny, because most banks keep your money in other peoples wallets. Irony!)