That means the weight of stones are perfectly balanced by the pressure
exerted by the gas.
Be careful. It is the pressure exerted by the stones plus atmospheric pressure that is balanced by the pressure of the gas. When you apply gas laws you use absolute pressure, not gauge pressure.
If the system is to do work by lifting the stones then the internal
pressure should increase and be greater than the weight of the stone
but this means the system is no longer an isobaric system
It is the external pressure that determines the work done by the gas. That pressure is initially constant (pressure of stones plus pressure of atmosphere) until you start removing stones in which case the external pressure drops. Now the process is not isobaric.
If you remove one small stone at a time the pressure of the gas will drop very slowly as well and always approximately equal the external pressure. But since the pressure is dropping, the process is not isobaric.
If you want the process to be isobaric, don't remove any stones. Instead, heat the gas so that it expands and does work raising the piston against the constant external pressure of the stones and atmosphere. Now the process will be isobaric insofar as the external pressure is concerned which is sufficient to calculate the work. If you want the process to be isobaric with respect to gas equilibrium pressure, then heat the gas very slowly so that the pressure of the gas is always only slightly greater than the external pressure. Then you have a reversible isobaric process.
so how actually is the work done by isobaric systems.
It is done by heating the gas very slowly while keeping the external pressure constant.
Hope this helps.