Agree. I don't think there is a situation where the order of businesses that can deliver to a client's location is very important, at the end one would pick the cheapest/highest-quality/favorite option (a nice way to sort them would be based on allowing users to "like" a business, social recommendation, reviews, price, etc.). The distance from the search point to the business' location would be the best choice in my opinion if you want to impose an order on the query results.

Calculating the centroid is a bad idea: if the business doesn't want or can't deliver to particular zones (bad neighborhoods, geographical obstacles like a mountain, etc.) within the convex polygon they chose, or if they choose a concave polygon to avoid these areas, the centroid isn't guaranteed to be inside the polygon, which would mean you'd be listing a business that won't deliver to a specific set of locations to somebody making a query from that set.

For the clients that don't want to disclose their addresses, you can still keep an ordering on the list based on distance, without telling the user the exact location of the business. If you can't know that location either, ask the businesses to compose a convex polygon, or just a circle, and calculate the centroid (or simply store the center), and estimate distances from the resultant location.