>The more they have, the more they can lend out based on the fractional reserve system.
Although the name fractional reserve might be correct it is highly misleading as to how modern banking actually functions. Practically speaking banks are limited by the availability of solvent debtors.
Colloquial language is describing an ancient process that is no longer in use. Debtors are people who promise to work, they sign a contract that they will pay in the future. That debt contract called a bond is an asset of the bank, it is a future income stream just like the share of a company. In exchange the debtor obtains a liquid form of his debt contract. The bank grants credit equal to the principal owed in the contract.
Essentially, bank money is just a share in the bonds that the bank owns. It's almost equivalent to buying a bond ETF whose share price is fixed to $1. The profits of the bonds are then distributed through interest rates. Now, the big question is, what happens when the bank is losing money because the bonds are worthless (think 2008)? It does not pass on the losses because of the free deposit insurance program that the government pays for. In fact, the incentive is to write as many (possibly bad) loans as the bank can get away with. Add onto the fact that US dollars are a shared currency among private banks, you get a pee in the pool scenario...
The central bank then engages in QE which is basically the act of transferring the bonds to the central bank and now private banks have a metaphorical share in the bonds the central bank owns. If there are 10 banks and 1 with 50% bad loans then after QE there are 10 banks with 5% bad loans. The central bank has the pee now and diluted it to the point the pee is no longer noticeable.
Of course, the "correct" solution is to just lower the value credit. That's usually done via inflation but the problem is that this "after the fact" inflation must be done through government spending. Wouldn't it make more sense to keep the value of credit the same just pass on losses directly? That would imply negative interest rates. In practice it doesn't work because cash gives a risk free 0% return which forces returns on every investment to be above 0% to justify the investment and that includes 0% interest rates on bank accounts. Essentially, inflation targeting only exists to allow representation of negative real interest rates on cash. It's a huge hack.
I wanted to limit it to the first paragraph but I felt it would be too difficult to understand without further explanation.