Would it be more insightful to teach students Lagrangian/Hamiltonian mechanics before Newtonian mechanics? What benefits would it bring to teach analytical mechanics before Newtonian (vector) mechanics? I began thinking in this after I saw a 2019 article in the magazine Physics Today that advocates teaching the concept of energy before the concept of force. I later thought that if it's better to teach energy before force, then it would be better to teach Lagrangian/Hamiltonian mechanics before Newtonian mechanics, since the former two are energy-based and only use scalars, whereas the latter is force-based and uses vectors. The only issue I could find is that high school and college freshman students would not have enough mathematical background.
 A: The concept of energy comes with luggage.
Potential energy does not have an intrinsic zero point. If you have two position coordinates then what is definable is the potential difference between the two.
(Incidentally, the fact that there is no intrinsic zero point of potential energy is what allows the convention to set the potential to zero at infinity in the case of the inverse square force law of gravity. You can put the zero point anywhere, as long as you apply that choice consistently. As an object moves from infinity towards a source of gravity the nominal potential energy of the object becomes more and more negative.)
Potential energy is well defined only when the integral of force from start position coordinate to end posistion coordinate is independent of how you move from start to end. A force law with that kind of property is called a 'conservative force'.
In all: in order to introduce the concept of potential energy you have to provide lot of contextual information, and you have to demarcate the classes of cases where the concept of potential energy doesn't apply.

Force is what you experience physically.
One example of a standard classroom demonstration setup is an air track. One way to impart velocity to a glider is to release the force of a compressed spring, for example a setup with a coil spring.
Example: two gliders, pushed away from each other by release of a coil spring.
You can try to squeeze the coil spring with your fingers, to get a feel for how stiff the spring is. To calibrate how much force the spring exerts as a function of the amount of compression you set up comparison with gravity; more compression requires more load.
The concept of force, when introduced in terms of physical experience, is straighforward. What you see is what you get.
In any education:
Start with the concrete, start with direct physical experience. With that established as a basis, proceed to introduce abstraction.
