Why does the use of a heavy, moveable piston ensure that any gas process will be isobaric?
The piston does not have to be “heavy” (a relative term, anyway). It simply has to have mass. If it has mass, $m$, it exerts a downward force of $mg$. If that force is uniformly distributed over a surface A, the force per unit area will be $\frac{mg}{A}$. If this weight is placed on top of a gas, the force per unit area is called pressure. If the piston/cylinder is surrounded by the atmosphere, the total external pressure is $\frac{mg}{A}$ + 1 atm.
A process is designated isobaric if the externally applied pressure is constant. For the piston that external pressure does not change, whether the piston is sitting on top of the gas in equilibrium, is accelerating due to a pressure differential, or is reversibly compressing or expanding the gas due to very slowly transferring heat out of or into the gas, respectively, with the surroundings.
Although the external pressure is constant, the pressure of the gas may not be constant except for reversible processes. For irreversible processes, temperature and /or pressure differentials may exist in the gas. Consequently, the ideal gas law would not apply to irreversible processes.
A mechanics of materials analogy to the above is a column subjected to a downward axial load (force). If the force is uniformly distributed over the cross sectional area of the column, the downward force per unit area is called normal compressive stress, $σ_N$. This external stress is constant; provided any axial deformation does not change the area over which the force acts. This is analogous to the piston surfaces and cylinder walls not expanding or contracting during the compression or expansion of the gas, respectively.
Hope this helps.