If there is a wheelie bin system, bags aren't usually collected. This is because the collection contract is specified on waste volumes from a known number of wheelie bins and their sizes. The rounds are then calculated and tested for a given number of vehicles and crews.
If the crews collect random numbers of bags as well, the vehicle may be full before it completes its round, plus they slow the crew down on their round which has all sorts of knock-on effects. As a vehicle can be faced with anything up to a 100 mile round trip to empty and return, there is no easy way of making that up without an extra crew day and that assumes there are crew to do it. The cost of extra crewing and even a single extra vehicle in the fleet will add scores of thousands of pounds per year to the contract, which we end up paying for. There's always a percentage overcapacity per round but that ends up getting squeezed each year as pressure on costs increase.
The other thing about bags is that handling them causes many more injuries to crew than using wheelies (another cost factor as much as a human one), the crew need sharps protection and that the bags leak and make it more unpleasant for all concerned.
No, I didn't manage a refuse service but a colleague did. It caused him a lot of grief!