I consider the energy-first approach as more natural. Energy is a scalar and using energy first, we can delay teaching vectors for solving 2-D problems a little longer, (albeit probably nor more than a few weeks at most). The harmonic oscillator equation from energy conservation is a standard integral treated is a good high school calculus class rather than to "guess" a solution to the differential equation. I have often thought when I was learning this > 40 years ago, suppose you are a bad guesser.
Given that my teaching experience has always been a TA and not instructor for the course, I never was given the autonomy for rearranging the lesson plan. I noted many years ago, I used a calculus textbook that started with integration rather than differentiation. Now that felt unnatural.