Does energy conservation imply time invariance?

It is actually the other way around. Time translation symmetry refers to Energy conservation. We define the Hamiltonian as $$normalsize {H} = Large{Sigma_i}normalsize{p_ioverset{.}{q_i} - L} $$ This says that the Hamiltonian in other words the energy is conserved when the Lagrangian has no explicit time dependance. i.e. $$frac{dH}{dt}=frac{partial L}{partial t}$$

This means as long as the laws of motion are time translation invariant, the energy of the system in consideration is conserved.

The converse is true as well. As you can see the equations say that

$$frac{dH}{dt}=frac{partial L}{partial t}$$

Which means that if energy is conserved it means that the Lagrangian has no explicit time dependance. Now even though it might seem in certain systems that the energy is not conserved, we must remember that the system is not necessarily isolated, so when we see that the energy is not conserved it just means that the flow of energy is from the surroundings