Your analogy is bogus. Fraud is fraud because it is intentionally dishonest and deprives a party of value. Fractional lending is sustainable because one, there's an actual exchange of value; and two, the majority of those obtaining the loans are faithful about paying them back under the terms of the loan, so you are willfully neglecting to include the on-going cash flow. If your argument is really that modern banking at the national level is a house of cards, then either limit the number of banks (i.e., the corporations, not the physical offices), increase the minimum capitalization requirements, or both. But let's not toss the baby out with the bathwater.
And again, at no point is there any "missing money" in the process. For a signature loan like a credit card, the bank pays on the cardholders behalf, so whatever purchase the cardholder makes is paid for out of the banks' aggregate operating revenue (meaning all transactions to any one other bank are aggregated between the parties and only the net difference actually changes hands). The balance is the "risk" that the bank carries, and presumably responsibly so based against the creditworthiness of the cardholder. For long term secured loans, the bank owns the asset until the loan is cleared, so there's no missing value here either. So where is the money being "created"?
The problem you have in the UK is that the banks that can issue the money are also commercial lenders. Not so in the US, where the Federal Reserve Bank only deals with the government and the banks, not individuals or commercial entities like corporations, and only the Fed prints the money.
You also seem to be missing the point that, while money = value, there is more to value than just money. The currency is just a token, and the electronic money just a tally. The real value is in assets and accumulated value through investments, commercial transactions and gainful employment, as *represented* in money. The notion that there needs to be a physical unit of currency for each expression of value (or nearly so), is to limit the economy to the possibilities of the physical travel of the currency; and today's global economy is proof-positive that that viewpoint is neither advantageous nor necessary.