You could call it grazing but ruminant animals must graze to ensure enough calories since they only eat grass.  Cows have 4 stomach chambers to a human's one.  Humans can better digest animal protein and fat which is highly satiating unlike the carbohydrate, sugar, and incomplete protein of grass.  When properly fed with a human diet, people would not feel the need to "snack", "graze" or eat between meals.
"Gathering" would only happen during short, specific times of the year.  Wild berries only appear during late summer and only for a few weeks.  Our ancestor's consumption of them caused the needed increase in body fat to get through the winter when animals were more scarce to hunt.  This put us into a ketogenic state for most of the year.  Disease was extremely rare. Today we eat sugar, refined carbs, tropical fruits and industrialized grains 365 days a year!  Not much wonder we're chronically ill with diseases our great-grandparents had never heard of..😜