That effect kicks in at higher sugar concentrations, e.g. a 2:1 sugar:water simple syrup, which can have some shelf stability. I don’t think soft-serve mix would be even close.

You need for there to be little enough water in the solution that cells with permeable membranes will be desiccated through osmosis. This can be seen in a common household science experiment, where you remove the outer shell of an egg by leaving it in vinegar for a while, then submerge it in corn syrup to watch its further transformation.

Sugar itself is food. It's the fact that sugar draws water out of the medium that it retards some kinds of bacterial growth (not botulinum for example).