Well, I think there could be two reasons why the last expression can be particularly useful, but neither one is really applicable in this case. In the first instance, both expressions are exactly the same, you have just posted a proof, if you know the velocity of the particle but not its momentum the first expression is better, why should you first compute the momentum to compute the energy? The same is true the other way, if you know the momentum but not the velocity, use the other one...
Again, they are exactly the same thing!
Now, it is true that for more advanced physics one will prefer using the momentum instead of the velocity:
The first reason is that the concept of energy in most situations is related to a more powerful concept, the Hamiltonian. But the math behind this requires the Hamiltonian to be expressed as a function of the position and the momentum, not the velocity.
The second reason, more practical, is that for relativistic particles the velocities cannot be as large as you want, in appropriate units they cannot be larger than 1. Therefore if you need to specify the velocity of a particle you need to give numbers like 0.9999 or 0.99999, which are not very comfortable to work with. On the other hand, the momentum can be arbitrarily large and therefore saying 70MeV or 200MeV is much easier.