I think there are serious pedagogical problems with such an approach. Here is a good general rule for explaining any kind of math:
Skipping over the motivation doesn't make something easier to understand.
As math educators, our instinct is always to simplify the math that we are presenting as much as possible. We always want students to understand the core idea first, and then build from that idea towards an understanding of the details. When presenting a new topic, we always try to strip away anything that seems extraneous or distracting.
But motivation isn't extraneous, and it isn't distracting. Motivation is necessary for human beings to understand mathematics. It's not part of the logical structure of mathematics, but it's crucial for developing intuition about the subject. If you don't know what something is for, or why it was invented, then you don't really have any hope of understanding it.
The narrative that negative numbers don't have square roots, so we should try making some up, is the motivation for complex numbers. It's where they came from. It's the only way for them to make sense. Absent this motivation, complex numbers will come across as completely arbitrary, and therefore incomprehensible. Students will feel like there's something they're missing, and they will be right.
You raise some objections to presenting this narrative, on the grounds that it leads to "confusion". I disagree with the spirit of these objections, for the following reason:
A student who asks questions is not necessarily "confused".
Often, students need to understand a subject pretty well to be able to ask intelligent questions about it. When a student asks how something they just learned can be generalized, or how to tell whether something really works, it means that they already understand it well enough to think about it critically. The goal of education is not to leave students dumbfounded.
Let me respond to your individual objections:
For one thing, if we're being rigorous, it's nonsense. What precisely does "all the usual rules apply" mean? What is your proof that a structure obeying the above rules even exists?
These are excellent questions, and I think any good explanation of complex numbers should address these directly, at least to some extent. The definition of complex numbers highlights the fact that we need to codify the rules of algebra (leading, ultimately, to abstract algebra), and raises the question of whether it suffices to have axioms for a mathematical object, or whether we really need to construct it.
Ultimately, this discussion is a bit too sophisticated for students at this level, but it should be possible to say something about it. A student who realizes that such issues arise in mathematics understands things much better than a student who thinks mathematics is always cut-and-dried.
It makes the geometrical facts about complex numbers seem very mysterious.
They are very mysterious. It's one of the great mysteries of mathematics that complex numbers work so well, that they are so intimately related to geometry, and that they arise in so many places. How did we start with square roots of negative numbers and arrive at Euler's formula? Why is complex analysis so much simpler and more beautiful than real analysis?
If something truly is mysterious, it's ok for students to perceive it that way. Point out explicitly how amazing and unexpected the geometry is, given that we didn't start with something geometric at all.
It leads to this type of confusion. In the words of 19th century mathematician John T. Graves: "I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties. (...) If with your alchemy you can make three pounds of gold, why should you stop there?"
Again, this isn't confusion. The student who asked the question on Stack Exchange has grasped the fundamental technique of complex numbers, i.e. making up new number systems to do what we want, and wants to know how far it goes. Intellectually, they are doing as well as Graves, and they are on the conceptual path that eventually leads to abstract algebra.
A student who doesn't ask a question like this after learning about complex numbers hasn't really grasped them. If you manage to teach complex numbers in a way that doesn't prompt students to ask this question, that means that you've taught less, not more.
Of course, you won't have time to fully answer questions like this, but that's ok. Students at the high-school level sometimes get the feeling that mathematics is some kind of closed book, and this is a great opportunity to point out how much mathematics there really is in the world to explore. Graves' question is one of the most important questions in mathematics, the motivation for huge swaths of modern algebra, and in some sense it still hasn't been fully answered. It's not a question to avoid --- it's the kind of question you should go out of your way to mention!