This isn't your council's fault but the lack of a decent MRF (Material Recovery Facility) in your area. These are private companies and they dictate how they receive the waste which in turn dictates how your authority collects it.
This isn't 100% true, down here in West Sussex we have just built a MBT plant (1 billion ££'s contract) to take black bag waste and it will take out the food waste and other recycling and make fuel, this facility was built by the County Council (disposal authority) and supported by us District Councils (collection authority) the problem is it was started about a decade ago, when the waste composition looked very different, its delivered by a private company but commissioned and paid for by the Council, the same with the MRF here run by a private company but paid for by the County.
If you cant recycle loads of stuff its because somewhere down the line, it could have been 5/10/15 years ago the decision was made not too or to take a different route, we have had quite a few changes to our MRF paid for from the recycling credits (below all in one bin) but facility's cost so much money its hard to make large changes.
2005 Launch Paper, card, tin cans, plastic bottles
2008 Added glass, tetra pak, aerosol
2015 Added Pots, tubs and trays
I am not sure how far up the chain you are where you work, I am a senior manager so I get to sit on the groups now making the decisions, its great to see someone else in the industry if you want my work email shoot me a trust message.