This is how markets work, yes
Savers pull £80bn out of Britain’s four biggest banks in protest of dismal rates
Savers are either switching to rival savings providers or paying off rising mortgage bills
We’d not say in protest ourselves. Rather, someone’s offering them a better deal elsewhere. Either a better interest rate on their savings or perhaps zeroing out some of their debt is that better offer.
But this is how markets work. No one at all looks at the price they’re paying for supplies - which is what our deposits into a bank are to a bank - and works out what is fair. They pay the least they think they can get away with. The thing which limits that minimal getting away with it price is what other people are willing to pay for that same thing.
This is true of everything in a market. Wages are not - contrary to belief - determined by what your employer thinks is fair to pay you. Rather, by what the next, alternative, employer is willing to pay you. Your actual employer must at least match that number (plus conditions) to retain your efforts. And so too of steel - the steel price is not set by what Ford is willing to pay, the price to Ford is set by what Stellantis, Aston and GM are willing to offer.
The interest rate offered on your savings is not determined by Lloyds deciding what is fair. But how much Lloyds has to pay to get you to deposit there. Something determined by the interest available on other deposit accounts and, even, the negative return you’re getting on your borrowings.
Banks must have deposits of course - no, MMT is wrong on this. Look at the bank’s financial accounts sometime. Deposits plus capital always equals loans out. Banks must balance their books in this manner each and every day (at 4.30 pm in fact). Thus, if Lloyds (to continue with the one named example) suffers an outflow of deposits it must find some from somewhere else. By, say, changing the price it is willing to offer for them - the interest rate.
It is true that banks have not increased the interest they pay on deposits as fast as they’ve been raising what they charge for loans. Fortunately we also have a system to deal with this - the market. Leave it be and she’ll be fine.