A first reason is that if we just look at the component functions, it might be the case that reasonable people could disagree about whether they "should be" dependent or independent. For example, consider the spherical coordinate chart $f:\mathbb{R}^2\to\mathbb{R}^3$ given by $$ f(\phi,\theta)=(\sin(\phi)\cos(\theta),\sin(\phi)\sin(\theta),\cos(\phi)). $$ In a literal linear algebra sense, the three functions here are linearly independent, but having a "rank" of a map from $\mathbb{R}^2$ be three is already sort of weird. Also, a reasonable retort to the claim of "independence" here is that these three functions have relations between them, just not linear relations---if $f(\phi,\theta)=(a,b,c)$, then it will always be the case that $a^2+b^2+c^2=1$, because I picked my function to parametrize a sphere. But this is what I mean when I say that we can already start getting into arguments about whether these components are "independent" or not---it depends on what kinds of relations we're willing to accept.
This map is actually useful for investigating a little bit more, so let's keep going!
The rank, as usually defined, depends on the point---this particular map also shows that there's a good reason for this! If we're at e.g. $\phi=0$, then changing $\theta$ doesn't change the map at all which suggests that the map is acting like a "one-dimensional map" at this particular point---which is captured by the fact that the map has rank $1$ at any point of the form $(0,\theta)$! The same thing also happens at points of the form $(k\pi,\theta)$ for $k\in\mathbb{Z},\theta\in\mathbb{R}$.
Geometrically, this is the "coordinate singularity" caused by the fact that $\phi$ here is measuring the latitude of our point on the sphere, and $\theta$ our longitude---lines of constant $\phi$ are the flat circles formed by intersecting the sphere with planes $\{(x,y,z):z=c\}$, and lines of constant $\theta$ are great circles passing between the north and south poles of the sphere. So, when $\phi$ is an integer multiple of $\pi$, we are at either the north or the south pole, at which point it doesn't matter what our longitude is---we're just at the pole.
There are some other things we can say about the rank of a map, and why we want to define it in this way, but hopefully this example is enough to show why how the rank gives us a lot of information about how a function behaves differently near different points!
