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Surplus local food does not seem to affect Growth, only direct modifiers to Growth affects it.
Example:
https://steamcommunity.com/sharedfiles/filedetails/?id=3337931838
But I couldn't find the formula for the "Base Value".
Comparing my cities, it doesn't seem like food surplus affect that Base Value - it seems to be mostly based on city size, but not sure if affected by anything else.
Pretty much all sources of food improve growth, so basically yes more food more growth. Food itself is not directly related to growth, but a farm provides 3% growth to the city, most food boosting buildings also boost health which boosts growth, and most consumables which add food like feasts and whatnot also provide growth. If you want a city to grow fast, focus it on farms/food.
You are correct that almost everything that gives local food gives Growth too.
Wait till your city grows in population and then you go in a spiral of food shortages then you can proudly say that farm isn't required. Have you played the game long enough though?