How to motivate this approach on Special Relativity?

One of the most common approaches to Special Relativity is that based on the two Einstein's postulates:

The laws of Physics are invariant in every inertial reference frame

The speed of light in empty space is the same in every inertial reference frame

Now, although this leads to the Lorentz transformations and so on, there is another approach. The other approach I talk about basically consists of considering Special Relativity as a theory about the structure of space and time where we reformulate the way we view spacetime by modeling it as an affine space whose underlying vector space where the difference between points exists is the Minkowski Vector Space $\mathbb{R}^{1,3}$.

This vector space is simply $\mathbb{R}^4$ together with the metric $\eta$ with signature $(+,-,-,-)$, in other words, there's some basis $\{e_\mu\}$ on which

$$\eta = \eta_{\mu\nu}e^{\mu}\otimes e^{\nu}$$

Where $\{e^{\mu}\}$ is the dual basis and where $(\eta_{\mu \nu}) = \operatorname{diag}(1,-1,-1,-1)$. In that approach the Lorentz transformations are those which leave $\eta$ invariant, that is, the mapping $L$ such that

$$\eta(Lv,Lw)=\eta(v,w).$$

From this everything follows as it does from the postulates, but it's better suited for generalization and makes clear that we are in truth changing the structure of space and time.

My point is, the first approach is well motivated: we have laws from Electrodynamics that seems to be correct, but they aren't invariant in inertial reference frames which introduces the question about which frame they are formulated with respect to and at the same time we know light has a speed $c$ but we also don't know with respect to which frame. This gives rise to the postulates which can be motivated and the rest follows.

Now how can one motivate the second approach? I mean, just saying: "spacetime in Special Relativity is an affine space whose underlying vector space has that structure and where the interesting transformation for Physics are those leaving that tensor invariant" although quite satisfactory on mathematical grounds doesn't motivate from a physicist's viewpoint what we are really doing IMHO.

So, is there a way to motivate the second approach, where Special Relativity redefines the structure of space and time, as we motivate the first, and lay down explict relations between this new approach and the original postulates?

All of the above + absolute time implies precisely Galileo's relativity and that the Galilean group is the unique group of transformations possible. Relax absolute time, and the Galilean group becomes just one of a member of a family of possible groups, which are all Lorentz groups with different values of $c$.
I applaud your approach, because the "what's so special about the speed of light" or "what's light got to do with it anyway" (SR) is something that drives many teenage scientists batty. When I first heard about relativity at ten years old, that question drove me to distraction. And I have been asked it by people of that age more times than I can count in my 51 years. When I was 15, I had read all about Maxwell's equations and thought I knew the answer: I knew roughly Einstein's first approach to the problem. I therefore had the impression that SR is ALL ABOUT light and that light is very special. There's nothing wrong with seeing the first approach to this problem. But you do not need light to explain SR and in your proposed approach it turns things around and says "hey, we found an experimental example of something that transforms with a finite $c$ Lorentz transformation. So what does SR then say about this thing?". In other words, we use SR to study and understand light, not the other way around. I simply think that an approach like yours is a more fundamental, simpler and pedagogically better. I don't think Einstein would have minded in the least that our approach to teaching SR should move on after 110 years!