Of course, one should introduce the action principle in Lagrange and Hamiltonian form as soon as possible in the university curriculum. It's impossible to do at the high-school level of course. The most important general subject to be taught early on in physics are symmetry principles, and indeed this should be done in the very first theory lecture on classical mechanics starting with Newton and then introduce also special relativity, of which one approach is to ask, whether Newtonian spacetime is the only possibility to realize the special principle of relativity, i.e., the principle of inertia, and the answer of course is that also Minkowski spacetime is a possibility, and then it's a matter of experiment to answer the question which spacetime concept is better to describe Nature with the obvious answer that it's Minkowski spacetime and thus relativistic physics.
As innocent as it might look, indeed the notion of mass is a problem in this approach, because to really understand it from the symmetry group-theoretical point of view, you need quantum theory. It's also important to keep in mind when teaching classical physics that this is an approximation and QT is the "real thing". So one should be careful not to introduce fundamental concepts that are wrong from the viewpoint of QT, and a relativistic mass is a paradigmatic example for a bad concept, because it doesn't match in any consistent way with the symmetry paradigm as the overarching conceptual framework of all modern physics.
From the point of view of classical Newtonian mechanics mass is pretty enigmatic. It's merely a property describing inertia (in Newton's Lex I) and active and passive gravitational mass (Newton's universal gravitational interaction law). Not to get things wrong from the group-theoretical perspective all you can do, what's anyway natural within Newtonian physics: Just treat it as a scalar to be introduced in finding the Lagrangian of a single particle fulfilling all 10 conservation laws from spacetime geometry, and you end up with ##L=m \dot{\vec{x}}^2/2##, where ##m## is a scalar proportionality constant, which turns out to have the meaning of inertial mass when considering also closed systems of 2 and more particles or open systems with particles moving under the influence of "external forces". Newtonian gravity is not a fundamental law but must be empirically introduced leading to the amazing conclusion that all three notions of mass (i.e., inertial, active and passive gravitational masses) are the same (Newtonian equivalence principle).
In this way you have as fundamental concepts of Newtonian mechanics the spacetime symmetries (Galilei symmetry) with the corresponding 10 conservation laws via Noether's theorem. There's no way to derive the conservation of mass, and that's already telling. Nevertheless you need it as another empirical assumption when considering continuum mechanics, where together with an equation of state it's needed to close the equations of motion in addition to the dynamical equations following from the 10 space-time conservation laws.
Further progress in the question of mass is then possible in two ways, depending on the order physics is taught as a whole. In the traditional way the next step is special relativity, and there you can start, as mentioned above, with the question, which spacetime models are possible given the Lex I only. Then you find that not only the Galilei group (and Newtonian spacetime) but also the Poincare group (and Minkowskian spacetime) are possible realizations of this symmetry principle. The very same symmetry analysis for a single particle than naturally leads to the notion of invariant and only invariant mass, i.e., mass stays a scalar quantity, but there's no empirical mass-conservation law anymore, i.e., you can use some example of relativistic reactions of elementary particles like ##\mathcal{e}^+ + \mathcal{e}^- \rightarrow \mu^+ + \mu^-## that clearly doesn't conserve mass but only energy, momentum, and angular momentum. For continuum mechanics instead of mass conservation you need to argue with some other conservation law like electric charge or baryon number together with some equation of state to close the equations.
From the symmetry point of view it's very clear that there's not even a useful idea behind the introduction of a non-covariant quantity called "relativistic mass". It also becomes clear that inertia is not (only) due to mass but due to energy and, as it turns out in the continuum mechanical context, also in some sense momentum and stress. This already hints at drastic changes necessary to also describe gravity. As we all know, it leads to General Relativity, which by construction makes all three kinds of Newtonian masses the same and just to a scalar parameter describing an intrinsic property of matter (as do the other fundamental properties like the different charges of the fundamental interactions described by the Standard Model).
With QT the role of mass becomes clearer, using the symmetry concepts. Here one needs the idea of unitary ray representations, leading to the conclusion that in general a classical symmetry can be realized by analyzing the Lie algebras of the classical spacetime symmetry groups leading then inevitably to the covering group of the corresponding symmetry group, which leads to the conclusion that instead of the rotation group of euclidean 3-space one can consider its covering group SU(2) leading to another intrinsic property of matter, the spin, which can be both integer and half-integer.
Analyzing then the Galilei group further it turns out that the Lie algebra admits a non-trivial central charge, which turns out to be the mass, and interestingly the case of 0 mass corresponding to a unitary representation of the covering group of the classical Galilei group doesn't lead to a useful dynamical quantum theory. So one has to extend the classical Galilei algebra by mass as central charge, leading to a superselection rule excluding superpositions of states belonging to different values of (total) mass, which explains the empirical mass-conservation law within relativistic mechanics.
The same analysis of the proper orthochronous Poincare group for relativistic QT leads to the conclusion that there are no (nontrivial) central charges of it's Lie group and mass is just a Casimir operator of the algebra with no need for a superselection rule nor an additional conservation law for mass in accordance with the empricial observation that in the relativistic realm mass is not conserved. Again there's no hint for why one should introduce an artificial concept of "relativistic masses" at all.
In summary, from a modern point of view thus there's no convincing theoretical or empirical argument for the introduction of a relativistic mass. To my knowledge there's not even a useful heuristic argument to introduce it!
