This isn't correct. Banks can only lend out what they have. They can't manufacture leverage out of thin air. Leverage is a function of being lent money, not of lending out money.
The correct example is that the bank receives $1,000 in deposits from Alice and is allowed to lend out $900 of that. However, this means that the bank only has $100 cash on-hand so it cannot return Alice's $1,000, at least not immediately.
This does not mean that banks do not employ leverage, banks do borrow money, typically in the form of a preferred stock issuance. Just that they basically aren't allowed to in the example you gave. This is all laid out in bank financials. for example, for 2014 Wells Fargo had assets of $1.7 trillion and loans of $863 billion. The Deposit base is around $1.1 trillion.
So, $863 billion in loans on a deposit base of $1.1 trillion.
The mortgage crisis created a situation where loan losses exceeded the regulatory pad for many banks, and in several cases made them effectively insolvent. The Fed provided liquidity temporarily to give the banks time to become profitable again in order to be able to get back into compliance. Which most did. Most of those that did not, such as Washington Mutual, were either forced to be sold (at the beginning) or became desirable assets sold to other banks who were able to take over the deposit base without incurring losses to depositors. Most of the FDIC's losses (since recovered) occurred with smaller banks who had gone so deep into the red that they could not recover even with the extra few years the Fed gave them to become profitable again.
That's the reality. You don't have to like it, but people who deeply believe in bad information tend to wind up unhappy their entire lives when it turns out not to be true, over and over again. There's been a lot of that, too.