Sainsbury's has never used tray liners (or not in my area, anyway). They went straight from bags to nothing, and it is a pain in the proverbial.
The food is piled into pallets, and depending on the sense if the packer can have heavy items on top of fragile ones (so squashed bread and burst cream tubs), and even if the stuff is sensibly sorted some of the deliverers put the pallet on the pavement (as opposed to the step) and stand back, so I have to ask them to stack the pallets to stop me having to bend and stretch over and over.
I keep a trolley by the front door, and load heavy items into that, so that I just have to wheel it into the kitchen, and the rest goes into a giant cool bag that lives in the trolley between deliveries.
It works better for me than going to the supermarket, which I haven't done for years before the pandemic, but it is becoming a real nuisance. I appreciate that the safety of the drivers has to be paramount, and that not much that is reusable is likely to be safe just now, but there must be an easier way than this.
Also, I do get sick of hearing people going on about how they do their own shopping as though it is somehow virtuous. It's great if you have a convenient supermarket, have enough time on your hands, have a car to get there (or to click and collect), or are healthy enough to carry the shopping home and that is what you want to do, but the low-level sneering at those who use home delivery is unnecessary.