>Please explain the mechanism by which a "bank" is able to transfer to you an amount of BTC that they do not have wallet control over?
BTC that's in an offline wallet is like cash under the bed; and you're right that no-one is lending money that they have under the bed.
However, a significant number of BTC owners don't keep their money in their own wallets - they keep it at an exchange, which almost certainly commingles it with other owners in shared wallets.
You can certainly take a maximalist position here and assert that the exchange is the real owner, because they have the private keys. The fact of the matter is that the real world includes a difference between legal and beneficial ownership; and has done since well before the existence of fiat currencies - because it's useful.
That still is not the same thing. Coins in an exchange can only be traded for other coins in the same exchange - it's effectively KrakenBTC or BinanceBTC or [...]. Unlike fiat money in a bank, these exchange balances exist only in their walled gardens and cannot be exchanged for goods or services, effectively decoupling them from the total supply.
The total supply of bitcoin is the sum of all unspent transaction outputs on the Blockchain. Not what a bunch of exchanges tell their users they have.
>Unlike fiat money in a bank, these exchange balances exist only in their walled gardens and cannot be exchanged for goods or services, effectively decoupling them from the total supply.
These are demand deposits with the exchanges/shadow-banks. You can (at least with some products; others look more like time deposits) withdraw your deposit at any time. This seems basically the same as a bank account where you can withdraw money at any time to buy goods or services.
I visit my "bank". I deposit my "currency" in a demand deposit account. I receive regular interest payments. My "bank" lends my "currency" to other people, in exchange for interest; presumably making a profit on the net interest margin. I can return to my "bank" at any time to withdraw my "currency".
How is this not credit creation? How is, e.g., Binance not operating as an unregulated shadow-bank without any kind of deposit insurance?
Note that the money supply increases simply as a result of banks not holding full reserves against deposits. Whether the bank lends or not is not important, as far as the money supply is concerned. The same applies to crypto-currency exchanges, of course, which can inflate the circulating supply of bitcoins by lowering the reserve ratio, regardless of whether they engage in lending.
You have to convert your Exchange!BTC into BTC to spend it on goods or services. In other words, it has to hit the blockchain. If it is not on the blockchain, then what is being transacted is BTC in name only.
You need not convert your bank balance into cash to spend it on goods or services. The balance itself can be used directly.
If the bitcoins can be withdrawn at any time (i.e. without a notice of withdrawal), then these deposits are just like a current account, which means they are considered a component of the circulating supply of bitcoins (what we would call M1, if bitcoin was a currency).
BTC that's in an offline wallet is like cash under the bed; and you're right that no-one is lending money that they have under the bed.
However, a significant number of BTC owners don't keep their money in their own wallets - they keep it at an exchange, which almost certainly commingles it with other owners in shared wallets.
You can certainly take a maximalist position here and assert that the exchange is the real owner, because they have the private keys. The fact of the matter is that the real world includes a difference between legal and beneficial ownership; and has done since well before the existence of fiat currencies - because it's useful.