I don't know why the automated workers do this. Due to the imp. costs, you shouldn't let them work automatically.... as we know, workers rebuild improvements when there is a current need (low taxes, production).
Tribal Warriors don't use food as Peasant Militia. You may not mix it up with the food storage feature of PAE while units not only costs hammers, they cost food too. no matter which civic you have. at least human soldiers. ships don't. You can see it in the unit pedia, where units have food costs next to prod costs.
This shall reduce city growth because young men left the city for war.
the next big point of this PAE feature is: collect grain. the more grain you own, the less a city loses food for unit creation. this means: if you own no grain resource: a unit costs 50% prod costs in food. if you own all five grain resources (wheat -10%, oat -10%,...) a unit costs 0 food extra costs.
so it is a passive bonus if you are able to possess all five grain resources.