Note that KMC didn't say soap or detergent. Simple Green or any soap/detergent will need to be washed off thoroughly. That means water getting into the places you don't want it to be. And Simple Green has been accused of causing hydrogen embrittlement in chains.

Don't you have any kind of petroleum lube around your home. Even motor oil will saturate a shop rag and let you get the lube off the sides of the chain that is annoying you. Or mineral spirits, or WD-40 or many other things.

If you are going to wax your chain, then simply wiping off the sides of the chain is not the only thing you need to do. But there are several ways to skin that cat too.