> The Lagrangian is great for solving many problems. But it is definitely not more general than Newtonian mechanics
Sure it is.
It's true that for certain systems, you can derive one from the other so in that sense they might be "just different notations" for the same physics (I'm disregarding non-conservative forces, which aren't really considered in fundamental physics).
But the Lagrangian formalism really consists of two parts: Hamilton's principle and a choice (postulate/guess) of Lagrangian (or Lagrangian density).
When we say that Lagrangian formalism is more general than Newtonian mechanics, it means we can describe physics using the Lagrangian which we can't get to via Newton's laws.
For example, you would be hard-pressed to derive general relativity from Newton's laws, but if you start with the Einstein-Hilbert action, you can derive Einstein's field equations.
> A better example of a fundamental law would be action-reaction or something.
This is another example where the Lagrangian formalism is more general.
In this case, a translation-invariant Lagrangian implies conservation of (canonical) momentum.
But in more complex systems, Newton's third law might may fail when canonical momentum is still conserved (the prototypical example is the Lorentz force law).
> Using Lagrangians to derive it then begs the question "but why must electrons obey the Lagrangian?".
This isn't circular as much it is one less level of indirection.
It's like if you say the reason a dropped object accelerates to the Earth is gravity, you have just shifted the question to "why must gravity behave as an inverse squared law". You can go further and say that Newtonian gravity is not fundamental, but is an approximation of from general relativity, where the dropped object isn't really accelerating. Again, the question is shifted to "why is general relativity described by the Einstein field equations". At each level of reduction you describe one phenomenon by something more fundamental. That you don't have a further explanation doesn't logically preclude that it's more fundamental.
Sure it is. It's true that for certain systems, you can derive one from the other so in that sense they might be "just different notations" for the same physics (I'm disregarding non-conservative forces, which aren't really considered in fundamental physics). But the Lagrangian formalism really consists of two parts: Hamilton's principle and a choice (postulate/guess) of Lagrangian (or Lagrangian density). When we say that Lagrangian formalism is more general than Newtonian mechanics, it means we can describe physics using the Lagrangian which we can't get to via Newton's laws. For example, you would be hard-pressed to derive general relativity from Newton's laws, but if you start with the Einstein-Hilbert action, you can derive Einstein's field equations.
> A better example of a fundamental law would be action-reaction or something.
This is another example where the Lagrangian formalism is more general. In this case, a translation-invariant Lagrangian implies conservation of (canonical) momentum. But in more complex systems, Newton's third law might may fail when canonical momentum is still conserved (the prototypical example is the Lorentz force law).
> Using Lagrangians to derive it then begs the question "but why must electrons obey the Lagrangian?".
This isn't circular as much it is one less level of indirection. It's like if you say the reason a dropped object accelerates to the Earth is gravity, you have just shifted the question to "why must gravity behave as an inverse squared law". You can go further and say that Newtonian gravity is not fundamental, but is an approximation of from general relativity, where the dropped object isn't really accelerating. Again, the question is shifted to "why is general relativity described by the Einstein field equations". At each level of reduction you describe one phenomenon by something more fundamental. That you don't have a further explanation doesn't logically preclude that it's more fundamental.