But if some hydrogen bonds between molecules are destroyed then why is the kinetic energy of these particular molecules not increased and, consequently, temperature?
The idea that temperature is proportional to the average kinetic energy of molecules is exactly true for classical ideal gases, and an approximation for other systems.
The fundamental property of thermodynamic temperature is that when two systems are placed in intimate contact, there will be a spontaneous net transfer of heat from the warmer system to the cooler one, while net flow of heat in the opposite direction is not spontaneous. The reason for this spontaneity is that the cooler system will gain more entropy from receiving a small increment $\delta q$ of heat than the warner system will lose from giving up the same amount of heat. This transfer of heat will continue until the two systems reach thermal equilibrium. At that point, their temperatures are the same.
For example, a litre of water at 50 °C is seen to be hotter than a litre of water at 40 °C because heat would spontaneously flow from the former to the latter and not the other way around. This would eventually result in both systems settling down at the intermediate temperature of 45 °C.
But when water is boiling, there is no state in between water and steam. Steam that is right at the boiling point does not spontaneously transfer net heat to water that is right at the boiling point. Any small increment $\delta q$ of heat transferred from the steam to the water would simply result in a small amount of steam $\frac{\delta q}{\Delta H_{vap}}$ turning into water and exactly the same amount of water turning into steam, so the entropy of the system as a whole wouldn't change.
For this reason, we must admit that the temperature of steam at the boiling point is the same as the temperature of water at the boiling point, even though the average kinetic energy might be higher in the gaseous state.