Pork cheeks would be neither terribly fatty nor chewy - they go very tender. You want to cook in apple juice/cider, and then you can serve the meat and leave the liquid. Once the liquid cools, the fat can be skimmed off (and, in my house, used to fry things!), and you have a lovely apple-flavoured pork stock to make soup with (it's really good with lentils or parsnips).
I don't personally find chicken is very cheap, especially whole chickens. You can't make decent stock from a cheapie one, and free range are expensive. Aldi does a half-decent free range chicken for around a fiver, and that's the best compromise I've come across.
Do you like things like frankfurters? Very cheap, and flavourful. They are fatty, but IME children tend to like them. I also really rate salmon trimmings - you can make pasta or fishcakes.
If you have a good butcher game can sometimes be cheap (it can also be eye-wateringly expensive, because it is both intrinsically cheap and slightly snob, so it does depend where you re in the country).
We were down to one income recently and I so much missed lamb - there really are no cheap cuts (only cheaper ones), and we'd have beef burgers on occasion, but basically red meat was out. The thing I did find was that all supermarkets will do a loss leader on some kind of meat at some point, and then I'd corner the market and freeze it all. I really think buying what's coincidentally cheap is (depressingly) more likely to make a difference than choosing different cuts.