Does Cauchy's theorem's hold if we only assume boundedness?

It's true that, as the square size approaches zero, the contributions from the individual squares tend to zero. But as the square size approaches zero, the number of squares tends towards infinity! For your argument to work, you need some sort of guarantee that the rate at which the integrals on the individual squares are getting smaller is sufficient to offset the rate at which the number of squares is getting bigger. This is where we require holomorphicity, rather than just boundedness.

For a different perspective on this, let's try and construct a rigorous proof in the special case where the original contour $C$ is one big square of width $L$. Although this isn't the most general case, it's good enough for the purposes of identifying where the proof breaks down.

So let's assume $$ \left| \oint_{C} dz \ f(z) \right| = \epsilon.$$ for some positive $\epsilon > 0$. We want to derive a contradiction from this assumption.

Well, if the modulus of the integral of $f$ around $C$ is $\epsilon$, then, subdividing the big square into quarters to create four mini-squares of width $L/2$, we see that the modulus of the integral of $f$ on at least one of these four squares must be greater than or equal to $\epsilon / 4$. Continuing subdividing in this way, we see that, for any $n \geq 0$, there exists some square $C_n$ of width $L/2^n$ such that

$$ \left| \oint_{C_n} dz \ f(z) \right| \geq \frac{\epsilon}{4^n}.$$

Does this lead to a contradiction? Not if we only assume that $f$ is bounded. With $|f(z) | \leq M$ (say), the ML inequality gives

$$ \left| \oint_{C_n} dz \ f(z)\right| \leq 4 \times \frac{L}{2^n} \times M,$$ and there is no contradiction. Yes, the ML inequality tells us that $\oint_{C_n} dz \ f(z) \to 0$ as $n \to \infty$. But $\oint_{C_n} dz \ f(z)$ does not get small at a fast enough rate so as to contradict $ \left| \oint_{C_n} dz \ f(z) \right|$ being greater than or equal to $\epsilon / 4^n$.

If we make the stronger assumption that $f$ is holomorphic, then we can get a tighter bound on the sizes of these integrals, one which does lead to a contradiction. Let $a$ be the (unique) point contained inside all of the squares $C_n$. Since $f$ is differentiable at $a$, we have

$$ f(z) = f(a) + f'(a)(z - a) + v_a(z)(z - a),$$

where $v_a(z)$ is some function that tends to zero as $z \to a$. There must therefore exist some $n \geq 0$ such that $|v_a(z)| < \frac \epsilon {8L^2}$ on $C_n$, and hence $|v_a (z) (z - a)| < \frac \epsilon {8L^2} \times \frac{2L}{2^n}$ on $C_n$. Applying the ML inequality now, and remembering that the integral of the linear function $f(a) + f'(a)(z - a)$ around any closed contour is zero, we get the tighter bound,

$$ \left| \oint dz \ f(z) \right| = \left| \oint dz \ v_a(z)(z - a)\right| < 4 \times \frac{L}{2^n} \times \frac \epsilon {8L^2} \times \frac{2L}{2^n} = \frac{\epsilon}{4^n}.$$

This bound, obtained by assuming holomorphicity, is much tighter than the bound we got from assuming boundedness alone, and it is tight enough to derive a contradiction.

Kenny Wong has nicely explained why your proposed argument is wrong. The conclusion of the argument is also false, and in fact is false in the strongest possible way: the result of Cauchy's theorem is true only for analytic functions and no others. More precisely, the following theorem (known as Morera's theorem) is true:

Theorem: Suppose $U$ is an open subset of $\mathbb{C}$ and $f:U\to\mathbb{C}$ is a continuous function such that $\int_\gamma f(z)\,dz = 0$ for any rectifiable (or even just piecewise linear) closed curve $\gamma$ in $U$. Then $f$ is analytic.

To sketch the proof, if $f$ satisfies this condition then we can find an antiderivative of $f$ by integrating (just like we do with the fundamental theorem of calculus for functions on $\mathbb{R}$, except that we need to know the integral on a closed curve is always $0$ to be sure that this is well-defined). But the derivative of an analytic function is analytic, so this implies $f$ is analytic.