What exactly do you mean by processed sugar?

Sugar can refer to any sweet-tasting carbohydrate, but most commonly the word refers to glucose, sucrose or fructose. There are bacteria which can consume all of these. “Processing”, at most, might break down glucose into the other two, but that’s not a significant change.

Nothing can survive in pure anything. Any substance is lethal if an organism is entirely immersed in it. Sugars or starches are a bacterial favourite, but yes, they too kill in high enough concentrations.

But not because they are intrinsically antibacterial. You’re not wrong to say that sugar is a preservative, but it’s still just sugar. The difference is in how much of it there is.