I think you are misunderstanding fractional lending and collateral.
Fractional lending seems a bit odd, but it's not like there is 'no collateral' - rather, there ends up being 'partial collateral'.
As it turns out, that 'partial collateral' is enough - it's actually reasonable thing to do, because the vast majority of loans are repaid, it's not necessary to fully collateralize everything on the whole.
Banks have to have some capital requirements, which means, if they 'screw up' too badly, then they go bankrupt! So they are acting in a capitalist manner and have to be careful about how they lend, and at what rates. If random bank acts irresponsibly, then random bank will go kaput by regular market forces.
Instead of thinking of fractional lending as 'missing money' think of it more like leverage.
Basically - the effect of fractional lending is that it's a 'multiplier' to whatever the Central Bank decides to do.
So it exacerbates effects in one direction or the other but on the whole, it doesn't change the real nature of the system.
The way we manage money is sound.
We need money to expand and contract, and we have good ways to measure that.
The 'danger' of fiat of course is political intervention, or a failure of controls.
We had a broad intervention in 2008 that favoured home owners over others, and there are attempts to use the Central Bank to do 'Social Justice' type things, which I think is very risky.
It's a bit like Nuclear Energy: it's very potent, you just have to watch it responsibly.
Finally - currency should be a 'current' asset, not a store of value. We just want it as a medium of exchange. So as long as it's not shifting too much one way or another, then it should work.
For 'stores of value' there are other things, like real estate etc..
"Inflation happens when the government prints money that has no collateral, and has no correspondence to added value in the economy and so it dilutes the value of the money that is already in circulation. "
This is misleading.
The 'government' does not print money, the Central Bank does. And as you indicate, the bank ultimately creates credit via fractional lending.
'Inflation' happens when the cost of goods rise faster than the money in circulation.
Thus 'inflation' can happen because 'stuff is more costly to make'. Like gas in Europe, is more expensive, not because 'money printing' but 'Russia'.
Also, inflation can feasibly happen without money printing or even a rise in inherent cost of goods, if the economy shifts in a way that ends up in excess cash.
But ultimately, if during normal course of 'growth' if there is no expansion in the monetary base, then you'll have deflation, which has bad externalizes.
So the goal then is to adjust interest rates / print or extract money from the economy so that prices stay roughly flat, with just a tiny bit of inflation. That is a dynamic process, not a static process.