Well I managed to get some cities to size 6 just after I hit the Classical Era - one thing I found helped massively was that I had the rabbit resource available so built Cree culture and sent out 15 Cree Trackers hunting - with the new butchery button for subdued animals (which btw is absoblinkinglutely brilliant, thank you!) I was sending several food back to at least one of my cities per turn and all those one-off +1 or +2 foods make a big difference over the course of a couple of hundred years! Plus I was bringing back a lot of herd type animals that I subdued near my borders to build their herd buildings.
Ah yes. Early game (prehistoric especially) well over 50% of my food and hammers comes from hunting. I forgot to mention that. BTW, I intend to try to track down the display inconsistency in the food next. If you could post a save game that has such an inconsistency (I have one in my game but it seems to have a different source than your image) that would be appreciated.