First of all we are not in a capitalistic society, we are in a corporate society, Big difference.
Secondly I'd simply argue that Capitalism or corpratism having full power on an economy is unstable due to creating bubbles which adore to pop.
Thirdly Banks as a Non profit service simply alters where the money goes, less money on bank fees... more money to buying products. The idea is to make money-holders not a source of instability to compensate for the mistakes of other environments and reducing the average debt by reducing the profit in sending out loans, reducing interest.
If you regulate banks to become none profit organisations, you will have no banks. The government would have to create a nationalised banking service and they're hardly more trustworthy than the banks.
Being owned by the banking system, it's not something that will happen in the UK any time soon. In fact that last couple of banks the government re-privatized were all loss makers, the taxpayer yet again being forced to take a hit for the banking systems benefit.
If the political class weren't tools, there are several ways some control could be brought back to the banking system.
The biggest would be to split up the investment and retail side of banking, so that the gambling side of banking cannot access people's bank deposits. Plus, and this is a biggie. When the investment banks bets blow up, people's deposits are safe. So the bank would be allowed to fail without taking down the entire monetary system. The other significant change would be to increase interest rates to sane levels, which would force banks to look to sound investments rather than using cheap money to make bets on the markets. This would cause some damage to the public in the form of mortgage defaults, but those mortgages were unsustainable in the first place. But the real beneficiary of super low interest rates is the biggest debtor of them all, government. Which is using them to lend itself money at record low rates, which they spend on god knows what; because the public aren't seeing any of it with all these cuts.
Growth focused economies are bound to always fail simply because growth is never infinite. So the question is to how to make the growth as sustainable as possible.
Growth is an inevitability as the population increases. But the primary driver for the need of growth isn't population, it's our money from debt monetary system. Stop creating money as interest bearing debt and you eliminate a significant portion of the need for growth. Though it's easier said than done, it's a fight that has gone on for centuries. America was the most recent attempt to escape this system, but of course it got them in the end when the Fed was established.