I am not aware of any state-sponsored teaching systems that "focus on understanding the essence of concepts". Among the widely known systems, Montessori methodology is pretty close to what you are asking about. That is, at least for math up to the elementary school levels - I have not followed it for later years. There are numerous articles online on the Montessori method, and Montessori math in particular. For example:
Montessori Basics: How Math Progresses Through the Levels — Hollis Montessori School: https://hollismontessori.org/blog/2018/3/19/montessori-basics-how-math-progresses-through-the-levels
Much of the Montessori curriculum is based on giving children exposure
to concrete materials first, then giving them incremental
opportunities to work to more abstract concepts. This is no different
when it comes to math.
What do we mean by concrete? The children are able to hold a material
in their hands. The materials are symbolic or representative of
something else (a number, perhaps), and that symbolism changes over
time until children are ready to let go of the materials and find
solutions on paper or even in their heads.
There are also individual teachers who use their own methods.
For elementary school math, there is Jane Kats, a well-known teacher and author of the popular Mousematics series of books. Jane Kats uses play-based methods and sensory materials extensively. Unlike the Montessori methodology, where there is more structure, order and only a limited set of (often expensive) specific activities/materials per concept, Jane Kats uses many activities/materials per concept. These include tons of games, such as board games, fun problems, and sensory materials that are either inexpensive to buy, or can be put together by any parent out of paper, tape, ice cream sticks, thread, and the like.
For high school physics, including Newton's laws and much else, I can only recommend from personal experience a simplified (but not too simplified!) approach based on college-level physics, with elements of The Feynman Lectures on Physics. The main feature of this approach is derivation of many of the laws of physics from a handful of fundamental concepts. This makes physics easier to grasp and remember, and also much more elegant. Needless to say, it requires a bit of math knowledge. This approach is based on the individual teacher methodology, and not widely used, AFAIK.
Montessori Math Lesson - Number Rods - YouTube: https://www.youtube.com/watch?v=fCHh1XP3pR0
Montessori Mathematics - YouTube: https://www.youtube.com/watch?v=wKSRCeyR5Ck
Math should be felt with your hands: Interview with Jane Kats | Musings of a Mathematical Mom: https://aofradkin.wordpress.com/2016/10/29/math-should-be-felt-with-your-hands-interview-with-jane-kats/