I think you kind of need to define a bit what fast food means and what you consider small.
Following the McDonald’s model with a drive through, self-seating dining area, no table service, etc. those are mostly going to be chains, although some chains are pretty small, only operating in certain areas with a handful of locations.
And there are exceptions of course, some independent restaurants do have a drive through (often I see this setup when they move into a location that was formerly a corporate fast food place) and there’s chains or locations that don’t have a drive through, or do have table service, etc. but those tend to be exceptions rather than the rule.
If you expand it a bit to include other small, no-frills restaurants with a heavy emphasis on fast service and take-out, there’s a lot of places that could fit the bill. Pretty much every town has some little hole in the wall burger joint, deli, sandwich shop, taqueria, Chinese takeout, pizzeria, bodega, snack bar, etc. that you could potentially argue fits the fast food category.
Most of those places probably don’t exactly advertise themselves as fast food and would probably want to avoid that label (although to be fair, the same can probably be said for most fast food restaurants, I’ve never heard a McDonald’s commercial call themselves “fast food”)
There’s also going to be some overlap with other categories, fast-casual, convenience stores, etc. where the lines get blurred about what can be considered “fast food.”
In general though, in America, the term “fast food” is usually going to refer to the larger chains, and the smaller independent restaurants with otherwise similar service will be called something else.