I do understand that it's done to "stimulate" the economy so people don't let their cash rot in a checking account, but isn't it a zero-sum game?

The issue isn't keeping money in a checking account, but under your mattress. Even in a checking account, your bank can still use a part of the money to create loans. Under your mattress, the money does nothing and that makes it more expensive for people who need money to pre-invest in something.

But if the economy always needs injections of liquidity because otherwise it fails, isn't the economy broken?

Kind of, but requiring cash on hand with no lending at all breaks the economy a lot more. It forces all investment to occur to only people who have money and it makes running out of money a much easier thing to happen. If you read accounts of eras where money was much harder to be lent, you'll see cash flow issues were much more significant even for large companies and corporations.