The E.M.F of a cell is the work done in moving a unit positive charge in a loop or from the terminal to the same terminal. The force it experiences is a conservative force. Therefore, the work-done in a loop should be zero which means the E.M.F of any cell is zero, which is not correct...What happens there?
Suppose you consider the popular analogy of a water pipe circuit with a pump supplying power and a turbine or something similar for the water to do work on. The water flows in a loop, so the net work should be zero as per your argument, but nevertheless work is done - the turbine turns.
The solution is that the water does work on the turbine but has work done on it by the pump. So the water is effectively transferring work from the pump to the turbine, and the net work done on the water is indeed zero.
The battery is a bit different because the battery supplies energy through a chemical reaction rather than a mechanical pump, so the concept of work is a bit vague here. Nevertheless the same principle applies. The electrons do work on the motor, or whatever they are connected to, then as they flow back round the circuit the battery does work on them. The net effect is to transfer energy from the battery to the motor and the net work on the electrons is zero as you say.
As BMS says in the comment, the EMF of the battery is the work done per unit charge on the current as if flows through the circuit, and this is equal and opposite to the work done on the current as it flows through the battery, so the net EMF round the whole circuit is indeed zero.
Any charge in the wire is experiencing a force which has two components: the force provided by the source (the battery in this case) and the force due to the electric field which develops in the wire to prevent the charge from accumulating and keep the flow smooth. The line integral of the total force is then the line integral of the two components. The electric field is conservative as you rightly pointed out and thus does not contribute anything to the line integral. The line integral of the total force ( The EMF, a very lousy term btw) becomes equal to the line integral of the force due to the electric field. This (EMF) is equal to the potential between the terminals of a battery in a resistance-less source
The EMF of a cell is not the work done in moving a unit charge around a loop, but the work done in moving a charge from one terminal of the cell to the other.
The sum of the EMF's around a closed loop is zero, since this is simply the conservation of energy - the work done by a conservative force in a closed path is always zero. This is Kirchhoff's second circuit law.