Why does "guess and verify technique" not involve circular reasoning?

Question: "Find an $x$ such that $x^2 = 9$."

Answer: "I magically intuited that $x=3$ might work. I plugged that into the equation, and found that indeed $x=3$ causes $x^2=9$. So my answer is $x=3$."

Or, to phrase it more formally: "Let $x=3$. Then $x^2 = 9$. So $x=3$ is my answer."

That's a guess-and-verify answer. The reason it's a valid answer is because all I need to do to demonstrate that I've found a valid $x$ is to display such an $x$ and prove that it has the right property. A proof need not say anything about how the creator came up with it.

Circular reasoning in this instance would go something like "Suppose $4^2 = 9$. Then $x=4$ would work. So $x=4$ is my answer." The key here is that we've got a hanging assumption at the beginning of the proof: we supposed $4^2 = 9$, and all the rest of the argument depends on that assumption, which we never proved to hold. The difference between this argument and the former, valid argument is that we're allowed to assign values to variables freely as long as we do so consistently (we can't let $x=3$ and $x=4$ in the same phrase), and that's what we did in the correct argument. But "suppose $4^2 = 9$" is attempting to assign values to other values, rather than to variables.

If you have a more specific and less obviously-trivial example, I'd be happy to walk through it.

I think there's another potential issue at play here. The "mysterious process" of finding the right guess $x$ often does involve making dubious assumptions. But all that doesn't matter, because it's not part of the proof. Once you have found the example, we don't care where it came from: it stands on its own, and one you prove (without any dubious assumptions) that it has the right properties, you're done.

Circular reasoning involves making an unjustified assumption; proof by guess-and-check, though, doesn't. What is (arguably) under-justified is the choice of example used; however, that doesn't make the proof invalid, merely more mysterious.

A proof is valid if it "follows the rules." There's no rule that says that the proof explain why each step is performed the way it is, the steps just have to be valid.