Well, you can repeat this argument and apply it to Galilean invariance. Start with a boost induced by changing inertial frame whose relative velocity is perpendicular to the original velocity vector and you end up with the same additive requirement above. Allow the relative velocity of the new frame to be in any direction and you get the same kind of affine relationship I alluded to when I mentioned non-orthogonal basis.

    v -> v + u
    without u ⟂ v
    |v+u|^2 = |v|^2  +  2 v•u  +  |u|^2
    K(v+u) = 1/2 m|v|^2  +  m v•u  +  1/2 m |u|^2
    

With the difference

    K(v+u) - K(v) = m v•u + 1/2 m|u|^2

that is traditionally derived with a Lagrangian approach to the problem.

Note that it also holds for polar coordinates (radial vs tangential components), complex coordinates, rotation matrices (RᵀR = I => (Rv)ᵀ(Rv) = vᵀv), etc...

> it doesn't hold relativistically

I understand that you allude to Einstein's theory –and I agree–, however it seems you also point at reality as the ultimate arbitrator as to what energy is exactly – and that's why you referred to the most accurate theory we have. Not false either, but this oversteps the idea that we can only make sense of energy as a measurable conserved quantity through the apparatus of our theories, that, in order to give a complete account of reality, must be compositional with respect to how the phenomena but also the mathematical tools combine, can be decomposed and recombined into something else without losing that invariant in the process.

Under this view, energy is something that emerges (not the thing itself, its invariance) under some kind of description closure. From within our theories, energy is invariant not just because the physical reality is. They are invariant also because the different perspectives they can shed on a phenomenon form a closure within a conversion graph. You can convert from cartesian to polar and back to cartesian coordinates and you will find the exact same conserved quantity. or from one inertial frame to another. This closure is what constitutes the domain of a theory: a graph that describes how to convert one perspective into another in order to make sense of another conversion graph, telling us how to convert from one energy type to the next (kinetic energy to position as potential energy in classical mechanics). You can't have one without having the other.

Conservation of energy is also conservation of the means of description, since energy is a locally exhaustive account of reality. And these local pieces must be glued additively otherwise there are holes or redundancy in the resulting composition since the invariance we call energy is inseparable from the closure of the transformations under which the theory identifies it.

I know close to nothing about relativity, but from the principles I sketched, I expect that I can shift rotation into a relativistic context and use the same additivity argument as above, and that the relationship between the energy we have from Newtonian mechanics and the one we gain in this new setting will be additive too. And this is what we have:

    Euclidean rotation becomes hyperbolic rotation
    Euclidean norm becomes Minkowski norm
    The energy-momentum invariant replaces kinetic energy taken in isolation
    Newton reappears as a term in the expansion

And in the equations 1/2 mv² composes with the rest via addition:

    E = γmc² = mc² + 1/2 mv² + 3/8 mv⁴/c² + ...
    S_rel = ∫ [ -mc² + 1/2 mv² + 1/8 mv⁴/c² + ... ] dt

How could it be any different? New energy and a new theory can only come as additive corrections to a previous account of energy if we want to conserve what it already explains. And under this new light any observed deviation from an old theory (for instance the precession of the perihelion of Mercury) naturally appears as an additive deviation from the original action functional (S_new = S_old + ΔS).

In fact, I think you can you can see the emergence of action, stationarity, and energy conservation just from a compositional bridge between two theories, T1 extending T0 with:

    - two functors between the theories:
      I: T0 -> T1
      R: T1 -> T0
    - that compose into an identity:
      R ∘ I = Id_T0

Indeed, if T1 can offer another way to decompose a phenomenon h,

    I(h) = b ∘ a

and we want to be able to recover T0 from the segments, not just globally once recomposed (i.e. there is no holes in our functorial bridge), then

   R(b ∘ a) = R(b) ∘ R(a) must hold too

And the only way to translate this into a composition-preserving scalar account is via addition:

   Z(b ∘ a) = Z(b) + Z(a).

Once this additive balance of composable conversions between T0 and T1 is made continuous (by the requirement that the bridge has no holes), you have action. Stationarity appears as the stability of internal gluing points ("∘" in R(b ∘ a) = R(b) ∘ R(a)), and energy follows from invariance with respect to temporal translation.

Energy as the Epistemic Invariant of Theoretical Extension. Mmmhh.