it is said that the temperature does not rise because the energy is
used to break the bonds rather than increase the temperature. Why is
this the case? Why can't energy be used to do both?
Energy can do both, but not necessarily at at the same time.
In your graph, where the water is being heated as as a liquid and where the water is heated as a gas, the heat transfer is causing an increase in the translational kinetic energy of the molecules and atoms, i.e. an increase in temperature. But it is not at this point a sufficient amount of energy to increase the separation of the molecules and atoms (increasing microscopic potential energy).
But when the phase changes occur (solid to liquid and liquid to gas) the heat transfer to the substance is increasing the separation of the molecules and atoms causing an increase in microscopic potential energy. At this time it is not causing an increase in the microscopic kinetic energy (increasing temperature). So the temperature does not change.
Hope this helps.