The genus of a connected surface can be defined as the maximum number of disjoint simple closed curves that can be removed from it without disconnecting it.
Why must the simple closed curves be disjoint? It seems that if the curves are allowed to intersect, the surface becomes even more easily disconnected - so the way to draw the maximum number of curves is to find disjoint curves anyway.
Doesn't this make the definition a little simpler? Or is there a way to draw $n$ curves with some mutual intersections that leave the surface connected, whereas any $n$ disjoint curves would disconnect the surface?
A torus $T = S^1 \times S^1$ has genus 1. But I can find 2 simple closed curves on $T$ which do not disconnect $T$, namely the curves $a = S^1 \times ($one point$)$ and $b = ($one point$) \times S^1$. Of course, $a$ and $b$ are not disjoint, so this does not contradict the definition of genus. This example shows that changing the definition in the way you suggest would produce a different numerical value for genus.