Why doesn't phase transition occur at temperatures different from melting and boiling points?

When you add heat energy to a material, the energy goes into all the modes of motion available to the structure: vibrational, rotational and etc. When a substance is in a particular phase, it is because the free energy of that phase is the lowest at that temperature. Only at the phase boundary are the free energies of the various available phases equal and therefore heat added moves the substance from the low temperature phase to the high temperature phase.

So, for example, as you heat water from room temperature to an temperature below 100C, the heat is going into the kinetic motion of the molecules, the various vibrational modes of the molecules and etc. But it remains as water because that phase has the lowest free energy in that temperature range. (Yes some molecules at the surface of the liquid are given enough of a kinetic energy push to send them into the vapor phase, but that does not mean the vapor phase has a similar free energy to that of the liquid.)

In a solid it is similar. When heating a lot of the heat energy goes into increasing the amplitude of vibration of the atoms or molecules around their equilibrium positions. When the amplitude reaches a certain fraction of the distance between these positions, the solid melts. In molecular crystals, the bonding is typically weak van der Waals and so they have low melting points. Metals have stronger bonding and higher melt points. Covalent crystals, have some of the highest melt points because of the strong covalent bond.

One final point. It is only at first order phase transitions that the temperature does not change as you move from one phase to another. You have to add a certain amount of heat, the latent heat, to make this transition. For second order phase transitions, there is no latent heat and you can move smoothly from one phase to another. If you are above the critical point of water on its phase diagram, you are in this region.

Energy that is input to a substance goes first to increasing the storage of internal energy in one of the three primary atomic/molecular modes available: lattice vibrations (phonons), intra-molecular translations (bending, stretching, and rotating of bonds), and particle motion. The first two are associated with bonds, either primary (ionic, covalent, or metallic) or secondary (dipole-related or hydrogen). When more internal energy is stored in the substance in one of the three modes, the temperature of the substance increases. The ratio between the amount of internal energy stored in a substance and the amount of its temperature increase is the definition of the heat capacity for the substance.

Consider melting transitions in general. Melting causes particles that are in bound spatial locations to break free and move freely. In melting, bonds associated with the first two modes break when the amount of energy stored in them is sufficient to overcome the constraints of the attractive bond potential energy that keeps the atoms/molecules together.

We have an additional constraint. Melting occurs only at a temperature where the Gibb's energy of the solid phase equals the Gibb's energy of the liquid phase.

Now consider this example. Suppose that we input a given amount of energy to a substance below its melting point. Suppose that the energy input is less than we need to bring the substance to its melting point and to melt the entire solid. Finally, suppose that we want that energy to contribute ONLY to breaking bonds (i.e. to cause the substance to melt, and apparently to cause it to do so below its true bulk melting temperature).

We would have to recognize that the energy input will have to be localized to a specific spatial region in the solid, not distributed throughout the entire solid. Why? Because we do not have enough energy to melt the entire solid, we only have enough to melt a portion of the solid. We will have to also recognize that, before any localized region can melt, it must still reach its melting temperature. Why? Because melting occurs only at a temperature where the solid and liquid phases are in phase equilibrium, not at any lower temperature.

We are left to realize that, we cannot input energy to a solid material and cause it to melt below its melting temperature. The energy must go first to increasing the internal energy of the solid to reach the phase equilibrium temperature with the liquid. Once the substance reaches it melting temperature, energy input causes the bonds to break. The substance melts.

A similar logic applies at boiling points.

One interesting caveat is worth further study. The melting points of bulk solids are defined values. A well-known principle is that melting transitions of solid surfaces generally start happening at effective temperatures comparable to around 2/3 of the bulk melting temperature. Why? Because the surface atoms/molecules have fewer bonds to break. You may find additional insights by investigating this phenomena further.