However, I have never done any movement/drill thinking of how I would strengthen my “core”. I understand that the field of “functional fitness” is a lot about core strength with many drills specifically targeting it, but from my perspective that is still an approach stuck in an isolation mindset. If you train natural movement (including postures)in a broad manner (running, jumping, climbing, lifting etc…) with proper/optimum techniques, you do not have to think of core muscles and strength: they strengthen without you having to think of it. When it comes to people are really out of shape, as a coach you want to have them perform the simplest and safest moves and incrementally increase challenge as you can observe their skill/condition building up step by step.
The only think you want to priority “target” in MovNat though is not body parts but skills. Body parts naturally develop according to how you develop your movement skills.