Dear Nicojo, since you mention philosophical reasons let me remark that, since a scheme $(X,\mathcal O_ X)$ is a locally ringed space, the most primitive concepts should be as close as possible to the data, the (generalized) functions encapsulated in the sheaf $\mathcal O_X$ .

At a point $x\in X$ , what could be more natural to consider as the COtangent Zariski space $\mathcal M/\mathcal M^2$ ? It just consists of the functions vanishing at $x$ modulo those vanishing at higher order. And the dimension $d$ of this space will already tell you if the (locally noetherian) scheme $X$ is regular or not at $x$, according as $d=dim\mathcal O_{X,x}$ or $d> dim\mathcal O_{X,x}$

In the relative case $X/S$ the sheaf $\Omega_{X/S}$ will give you a lot of information. Just its nullity at a point already tells you (under a mild finiteness condition) everything about nonramification:

$\Omega_{X/S,x}=0 \iff $ $f:X\to S$ is unramified at $x$

And this is only the beginning: by taking the top exterior product of $\Omega_{X/k}$ you get (in the smooth projective case over an algebraically closed field $k$) the canonical sheaf $\omega_{X/k}$ which is a key concept for Serre duality. This canonical sheaf also plays a fundamental role in the classification of curves, surfaces and higher dimensional varieties, arguably the very heart of classical algebraic geometry . For example to $X$ you associate its canonical ring $R$, a graded ring whose degree $m$ component is $R_m=\Gamma(X,\omega^m)$.
The Kodaira dimension of $X$ is $\kappa(X)=trdeg_k (R)-1$ and general varieties are defined as those with $\kappa(x)=dim(X)$. They are supposed to be generic in some sense and have been intensively studied, in particular by the Japanese school (Kodaira, Iitaka, Kawamata, Ueno, Mori, ... )