If you start from the energy equation
d/dt(sum_j m_jv_j^2/2) = F.V,
here F is the total external force, sum_j means summation over all particles with summation index j,
V is the center of mass velocity (vector).
and if you assume this is valid in any frame, for any system of particles, i.e.,
d/dt(sum_j m_j(v_j+V'_j)^2) = F.(V+V')
for arbitrary V' (vector),
then you can derive Newton's second law!
The trick is that "valid for arbitrary V'" contains the same amount equations as the Newton second law, which is a vector equation.
No wonder you can derive the conservation of momentum from the assumption of energy conservation in any frame!