Basically - councils have a monitoring role, so have to hassle businesses to find out who their provider is and when waste is collected. If they can’t get those details, they have investigate what the business is doing with their waste, and prosecute them if necessary.
They also still have a general duty to keep the streets clean, so if Bins-R-Us didn’t collect (and the business didn’t chase them up, or they did and Bins-R-Us just didn’t do anything), or the business just doesn’t use anyone, we’d just end up picking it up.
So much simpler just to not bother and make (and fund) councils to collect all waste.