Prove $D_x(\int_a^x f(t)dt) = f(x)$

Your first line has a serious problem: $\int f(t)dt$ is not a function of $t$; the indefinite integral is a family of functions, not a single function. So saying "Let $\int f(t)dt= F(t)$" does not make sense. So as soon as you try to get started, you are already in trouble.

Now, you can say, "well, pick some antiderivative instead of taking the whole family." But if you try to define $F$ as "pick some antiderivative of $f$", then your problem is that you have no way to guarantee that there is such a thing in the first place! The whole point of this result is showing that there is an antiderivative, so you cannot assume there is one to start with.

Of course, if you assume the First Fundamental Theorem of Calculus and that $f(t)$ has an antiderivative, then this result is very easy, exactly as you do it: $\int_a^x f(t)dt = F(x)-F(a)$, so the derivative with respect to $x$ is $F'(x)=f(x)$. But this assumes that there is an antiderivative for $f$ and that the FTC (Part 1) holds; but since this theorem is often used to prove Part 1, that could also make your argument circular.

But, really, the major flaw is that you are assuming that there is an antiderivative for $f$ in the first place (you can prove FTC (Part 1) without this result, so the circularity problem is not fatal).