The problem with "save in good times, spend in bad" is that people don't actually save in good times. Even if you raise interest rates, people will still borrow money when times are good because that's when there's profit to be made by investing the borrowed money. They do it less when rates are higher, but they still do it.
The "problem" with the natural cycle is that in bad times people start to default on their debts, which would otherwise cause banks to raise rates to account for the higher risk, which would cause even more defaults. By central banks lowering rates at those times you prevent that from happening, but at the cost of ballooning everyone's debt out of control. Because then people affected by the downturn can borrow even more money at the low rates. So we borrow in good times and borrow even more in bad times. Real debt per capita has been increasing for decades, which obviously isn't sustainable.
But that's what Keynesianism does -- it defers a crash by allowing otherwise uncreditworthy people to take on additional debt rather than filing for bankruptcy, but nobody ever pays the debt back.
Now the problem is times are good except that everybody has too much debt, which is only fine because interest rates are low. Raise them and everything falls apart as the money people are currently using to buy stuff and live their lives is suddenly required to make interest payments on existing debt.
What we need at this point is for people to pay off their debts. But they can't, because the money they borrowed is now inside corporations that never spend it.
So we need to either find a way for that money to get out of corporations and back into the hands of regular people (without crashing the stock market), or print a bunch of new money that isn't derived from debt and give that to regular people so they can use that to pay back their debts with.