Are “zero-bag fares” really profitable for airlines? Any extra weight costs the airline in fuel. So, if none of the passengers carry any extra luggage (be it checked in or cabin luggage), will it lead to lesser fuel for the airline, thus lesser expenses, which would justify the lower  “zero-bag fares”.

Or, do the airlines have to meet some minimum weight criteria anyway to make any feasible fuel savings.
That's mostly a marketing ploy; people that don't have any bags are more likely to go with a airline that has a no-bag fare.

However not having any checked-in luggage means less overhead in terms of baggage handling as it's a bag less to load and keep track of. It also means that if you got lost and didn't make it to the gate in time they don't have to offload any luggage but can simply pushback as soon as the gate closes.

The freed-up space is then available for non-passenger cargo transport like mail.