I have this struggle too, but more with things that are not worth sending to a charity, such as small cheap toys, I have tried zerowaste bags but they are expensive.
I do see that for people without cars it is much harder, my local hospice charity takes loads and has a drive up donation center.
My local tip has to be booked and you have to provide the vehicle registration you will use, and they have number plate recognition barriers in place, so makes it difficult for anyone without their own vehicle. Weekend slots can be difficult to get and may need to wait a week or 2 which is going to be difficult for households working Monday to Friday. (Weekday slots are easy to get).
Dunelm stores have collection bins for old household textiles including pillows and duvets, so that's helpful if you can get to such a store, but who wants to lug an old duvet on the bus?
Some skip companies advertise that they recycle large amounts of what they collect, so that maybe an option for households with a lot to get rid of, but then you need somewhere to put a skip which is not easy if you don't have a drive or can't easily park elsewhere if the skip is in your parking spot.
But on Batteries there is no excuse to put them in the household waste, with the associated fire risk. As mentioned already on this thread some councils collect them with the waste if you bag them separately, so check your council's website, if not then it should not be really difficult for most adults to manage to get to a shop that sells batteries and if they sell them they have to collect them for recycling.