I've never considered that a lie, because I've never encountered a bank the claims to be holding all the money. They merely claim that I will be able to withdraw my money, and that's on average true.
I consider that claim to be in the same category as the grocery store claiming I'll be able to buy toilet paper today. Sure, unless there's a pandemic. But usually? No pandemic.
I've been using a brokerage account for banking for years.
The online P2P lending boom here in the UK from financial crisis onward gave people this opportunity.
The end result, 10+ years in, has been subdued returns (relative to e.g. stocks) and crippling illiquidity, as you are exposed to all those late payments, defaults, and have no access to a functional secondary market.
Honestly you're lucky if you can get most people to even use a separate savings account. I know many people who keep 100% of their savings in their current (checking) account.
Last year you saw both that banks were 'forced' to withhold dividends and they were perfectly capitalised. Both cant be true of course.
If they hide it from you your money and everyones money is safe. If they tell the truth everyone loses, so it is actually the best outcome for you and everyone else. The fractional reserve system relies on not everyone withdrawing their money at once.