This book I checked out from the engineering library on SR says of its sketch of relativistic kinematics:
We must stress that what is required here is the judicious invention of axioms to be placed at the head of the new mechanics; there is no logically binding way to derive them
and proceeds vexingly to be like "well! hm! I guess maybe for the relativistic momentum I just need to add a gamma here? looks like it works!"
While I admit the statement as quoted is probably strictly correct, I wish they had chosen a nice set of axioms, at least, to derive consequences from. Why not just say, ok, let's demand that there's a (position and time independent) Lagrangian (which consequently is just a function L(v) of velocity) and it has to be Lorentz-invariant, in the sense of any integral of the Lagrangian over a path has to give the same answer as the integral over the Lorentz transform of that path. Then I look at a particle sitting still at the origin for one time-unit, and the integral of that history is L(0). Then I look at its Lorentz transform, and see that I get the answer γ L(v). So it must be that L(v) = L(0)/γ, and the exact same traditional definitions of momentum and mass in terms of the Lagrangian (as its derivative with respect to velocity, and the momentum divided by velocity, respectively) give you L(0)vγ and L(0)γ, correctly predicting relativistic mass increase. L(0) is of course now interpretable as the rest-mass of the particle.
I guess maybe arguably it's just as arbitrary to shunt around things precisely so that the traditional definitions work — but I think it serves as some small piece of evidence that Nature is "actually working in" the Lagrangian formulation or something similar, and that its notion of energy is primary, and that mass and momentum are derived concepts; namely, they're just the sorts of things that get conserved when your physics are (time- and space-) translation independent.