Why does temperature remain constant during a state change?
The general answer I find in most places is that during a state change, the energy supplied is used to change the potential energy of the molecules in the substance and not the kinetic energy.
On Quora, one user writes "When the solid has been completely melted into the liquid phase, adding more energy will once again increase the average kinetic energy of the particles (make them move faster)."
With this in mind, my main question is this:
Do all particles in the substance undergo the state transition at the same time?
Consider a solid to liquid state transition over an interval $0 \leq t \leq T$.
Individual particles in the substance will break bonds at different points in $[0, T]$.
Heat being input into the system, if distributed randomly, could therefore act to increase the average kinetic energy of these early transitioned particles since their bonds have already broken. Therefore, the average kinetic energy of the system will therefore also change (perhaps slowly, perhaps quickly, I'm not sure) over the interval $[0, T]$, which by definition means the temperature of the system will change.
Instead, however, we observe a very flat line on heating curves at a state transition as if to suggest (as stated by the Quora user), it is only when the whole solid has completely melted does the temperature start to increase.
So what happens to the particles that break their bonds early on in the interval $0 \leq t \leq T$? What dictates that the whole system should be completely melted before the temperature increases?
Hope my question makes some sense.