It's true that bank loans create money rather than banks acting as middlemen who match up depositors with creditors, but savings and loans are still fundamentally tied together. The reason banks can create money through loans is because the loaned amount becomes a balancing deposit in one of their accounts, and even if it's used to pay people that use other banks that's fine so long as the outflow of loan-created deposits to other banks is balanced out by inflow of deposits created by loans. If they charge below-market interest rates and everyone starts moving their money elsewhere, this money printing trick breaks down and the magic money starts turning into actual debt.

So even though banks aren't really lending out savings in the way people assume, they still need to keep their saving account returns competitive and somewhat related to money they're making on loans. The main difference between the incorrect model and how banks actually work has to do with the willingness of the banking industry as a whole to lend out money and the availability of credit - that genuinely is pretty much untethered from saving rates.

Yes, I understand that banks don't need savings to generate loans and the two aren't tied, and my answer didn't say that.

My point is that the market determines savings rates, and when interest rates are higher, banks can afford to give higher savings rates.