Cauchy's theorem: what about non-smooth homotopies?

As has already been pointed out by Akhil in the comments, any two smooth curves $\gamma_0, \; \gamma_1$ which are continuously homotopic are also smoothly homotopic. The point here is to approximate a continuous homotopy, rather than the curves themselves.

More concretely, given homotopic smooth curves $\gamma_0, \gamma_1: I \to \Omega$, let $H(s,t): I\times [0,1] \to \Omega$ be a continuous homotopy between $\gamma_0$ and $\gamma_1$.

By the approximation theorem of your choice (mine is due to Whitney), we can find a smooth map $G: I\times[0,1] \to \Omega$, which coincides with $H$ on $I \times \{0\} \cup I \times \{1\}$ (since $H$ is smooth there). This means that we find a smooth homotopy $G$ between $\gamma_0$ and $\gamma_1$, so we are done in this special case.

Now, if we start out with rectifiable curves rather than smooth ones, we can find two smooth curves $\tilde \gamma_0$ and $\tilde \gamma_1$, which approximate our initial $\gamma_0$ and $\gamma_1$, respectively, and such that the linear homotopy

$$H_\lambda(s,t) := t\gamma_\lambda(s) + (1-t)\tilde \gamma_\lambda(s) \qquad \lambda = 0, 1$$

maps into $\Omega$ (choose a $\epsilon$-neighborhood of the image $\gamma_\lambda(I)$ of $I$ which is contained in $\Omega$ and take $\tilde \gamma_\lambda$ to be contained within this neighborhood).

It is not difficult to see that $H_\lambda$ will then be a "rectifiable" homotopy.

But with $\tilde \gamma_0$, $\tilde \gamma_1$ we are again in the first situation, so there is a smooth homotopy between them. Now we can build a rectifiable homotopy between $\gamma_0$ and $\gamma_1$ in three steps

1. Homotop $\gamma_0$ to $\tilde \gamma_0$ by the linear homotopy.
2. Use a smooth homotopy between $\tilde \gamma_0$ and $\tilde \gamma_1$
3. Go from $\tilde \gamma_1$ to $\gamma_1$ by the straight line homotopy.

Thus proving that all notions of "homotopic" agree.

If I understand your question correctly, the problem that $H$ may be non-smooth can be solved by approximating with polygonal smooth paths, see for example Rudin's real and complex analysis (3rd edition), thm 10.40 and the remark after it.

As an interesting note, Rudin adds that another way to circumvent this difficulty is to extend the definition of index to closed curves, which is sketched in one of the exercises of the book.