I get that cash is basically an IOU. I also get that an IOU doesn't really require any token; a shake of hands is enough to establish a debt, which is the same as money.

So it seems that in an economy without money, money will arise automatically, in the form of debts that are to a greater or lesser extent formalised. So why do economies need central banks? What's wrong with that kind of "informal money"?

I have a strong suspicion that we don't; that it's governments that need central banks.

We created central banks after the organically grown system crashed spectacularly a couple of times.

This is generally the way we react to that sort of things. And even if things still sometimes break (in slightly different ways than before) these systems have allowed us enormous amounts of economic growth.

We don't need central banks as they are today. We need something like a demurrage currency, once you have that you can control the velocity of money, once the velocity of money is known, you can do price level targeting via monetarism aka no inflation or deflation.

The central bank won't have to set the interest rates anymore if the reserve requirement is something sane like 20% or 30% as banks are limited in their bank reserves and thus incentivized to choose the right interest rate for themselves. This also eliminates the need for QE.

You still need the central bank as an inter banking network by the way.