Why is degree of dissociation used to estimate strength of an acid?
I would answer only the constant of dissociation $K_a$, because it is a thermodynamical constant and you can calculate it from potentiometric measurements without dealing with activities and so on. If you have the $\Delta G$ of reaction, which is related with the difference of electric potential of the cell, you can calculate the constant as $\Delta G = -RT \ln K$. The pH is not an indicator of the acid strenght, but only its consequence. If you have a very diluite solution of sulfuric acid, its pH will be near to 7, but it does not mean that sulfuric acid is weak. Also the degree of dissociation can give you incorrect information if you are dealing with very diluite or very concentrate solutions.