While I'm talking about fat I ought to just run through the two rat experiment described by Daniel Quinn.
You place a breeding pair of rats in a very large cage with, let's say, 5kg of rat food. At the end of 24 hours you remove any left over food, put in a fresh 5kg and repeat the process ad infinitum. Remove the excess, never allow more than 5kg. Clean them out occasionally.
They will breed. Their children will breed. Their children's children will breed. The population will grow. Eventually there will come a day when there is no rat food spare at the end of 24 hours. The young rats will continue to grow, maybe some of the older ones will die, those in their prime may loose a little weight. That's life for rats in the cage.
Ultimately an equilibrium will be reached between 5kg of food and a dynamic population of rats. That population will not fluctuate wildly.
You cannot make new rats out of thin air. Older rats dying will free up some food supply so more youngsters will survive. It will balance out.
This was exactly the situation for the humans on Kitava when Lindeberg did his study. On an isolated island, using limited subsistence agriculture and no external power or fertiliser input, there must be a balance struck between population and food.
Back when everyone on the Earth either subsistence farmed or hunted/gathered the same situation must have applied. Hunting on the neighbouring tribe's patch might be highly unpopular and this would set limits on food availability, even with a full continent to use. People were everywhere that would support life.
I don't think that continuous hunger was ever the norm, but obesity would be unheard of. A sustained glut of food would produce extra sprogs to take up the slack. Hard times would be hard on the vulnerable. We humans did this successfully for millions of years. We are no different to rats. We can only breed to our food supply, but we always will breed to that supply.
There never was enough spare food for obesity until the Egyptians started with storable Spelt. It all went down hill from there.