Why does benzoyl peroxide homolytically cleave to give phenyl radical
The mistake you're making (which is an easy one to make) is that you're neglecting to consider the carbon dioxide product. Carbon dioxide is a stable molecule, and its release also increases the entropy of the system. 

So even if the phenyl radical is less stable than the benzoyl radical intermediate, the combination of phenyl radical + $\ce{CO2}$ can still have an overall lower free energy than the benzoyl radical.