[zozbot123]

From a feedback control perspective, it amounts to trying to balance an inherently unstable system. Think of a cart-and-pole apparatus where the top end of the "pole" (that is, the natural rate) is constantly being pushed around by unpredictable, outside shocks, and you have to move the "cart" (the policy rate) in the same direction to make up for those and keep the whole thing from falling over (into hyperinflation or extreme deflation - an illusory "boom" or a very real and persistent "bust"). The zero bound is only a leftwards boundary for the "cart", not the "pole" - but when it's hit, you do need something like QE to push the top end of the "pole" rightwards again independently of the "cart".

(The way out of the mess is to stop trying to use monetary policy to manipulate market interest rates, and to instead shift to a short-term policy target that fosters stability rather than instability. Such as, e.g. the money-price of gold. Or some measure of the money supply. Or the market forecast of nominal incomes x months in the future. There are lots of plausible choices!)