special relativity - How to motivate the importance of the spacetime interval

The spacetime interval is a rather important thing in Special Relativity. It allows us to define the separation between any two events as spacelike, timelike or lightlike and more importantly, the Lorentz transformations can be defined as the transformations which keep the spacetime interval fixed.

In that sense, as we know: lenghts and time intervals themselves are observer dependent. They are not absolute notions. On the other hand, the spacetime interval is one absolute notion.

Now, given events $(t_1,x_1,y_1,z_1)$ and $(t_2,x_2,y_2,z_2)$, its definitios is:

$$I = -c^2 \Delta t^2+\Delta x^2+\Delta y^2+\Delta z^2,$$

so that it is the difference between the distance that light traveled between the two events and the spatial separation of the events.

My problem here is that if we try to construct special relativity following the historical procedure there is a certain gap when introducing the spacetime interval.

We can usually go on like this: we start reviewing the problems in electrodynamics which motivated the theory as Einstein himself stated in his paper. After that, we can follow Einstein's procedure and starting with the postulates derive the relativity of simultaneity, the lengths contraction and the time dilation. From that we are able to get the Lorentz transformations.

The next natural step is to give more mathematical substance to this construction, and start labeling events with elements of $\mathbb{R}^4$ so that we finally get to the idea of spacetime. The problem is that in this point one usually just defines this formula for $I$, shows that it is preserved by the Lorentz transformations and shows that it allows us to classify the separation between events.

What I want is to be able to motivate why do we introduce the spacetime interval. It is just a certain object with certain properties, but how do we motivate its importance in the context of relativity and how do we motivate its definition?

After all, as far as I know, originally relativity is trying to solve the inconsistency between Newtonian Mechanics and Maxwell's Electrodynamics. The Lorentz transformations would seem to already to the job. How could one in this context motivate the definition of $I$?