Yes, there’s usually little fat on wild mammals (..) Apparently they get some fat when they have access to corn fields.
That's not necessarily true.
It depends on the species.
Some animals are more predisposed to getting fat while the others stay quite lean regardless of what they eat.
There are very fatty wild seals, walruses, polar bears and relatively lean caribous. And neither of them have got access to any corns whatsover.
In Africa there are fatty hippos and very lean antelopes; still no acorns.
Look at the Iberico Black Pigs -
http://tiny.pl/hct9tThey practically only eat grass and acorns. Quite fatty they are, arent they?
But give the same food to goats and they will remain very lean.
On the other hand the sheep on the same grazing could get heckuva fat.
And so on...