Doesn't exactly work like that. They're pretty prompt to get their legal team involved. If you don't pay it gets escalated but not to the people who would get your bins emptied or the road swept. It's basically a tax that goes with owning a property, not an invoice
If they're not doing what they should then report to the right dept, missed bin collection, uncut verge, broken streetlight etc
Also, the tax funds the council, so covers welfare, social care etc, and also streetlights or road repairs on other roads you drive on (you don't just pay for a streetlight or bit of road outside your house)
I believe police and fire brigade funding comes from it too. Also a chunk towards parish council.
So unfortunately, not paying will achieve nothing, phoning them and complaining may well do tho.
Our council gives a very clear rundown on what the money goes towards, other than social care it's stuff I get something for, or fire/police which I may need one day.