However, there are thousands of human languages, which have all been changing their pronunciation for (at least) tens of thousands of years in all kinds of ways, and they keep changing as we speak. If such a happy fixed point existed, don't you think that some of them would have already hit it by now?

No, I don't. Evolution is always a hack of what came before it, whereas scrapping the whole thing and starting from scratch doesn't suffer from that problem. I don't need to hack an existing structure; I can build exactly what I want right now.

Here's an excellent example of this general point: Self-segregating morphology. That's the language construction term for a sound system where the divisions between all the components (sentences, prefixes, roots, suffixes, and so on) are immediately obvious and unambiguous. Without understanding anything about the speech, you know the syntactical structure.

That's a pretty cool feature, right? It's easy to build that into an artificial language, and it certainly makes everything easier. It would be an important part of having a stable sound system. The words wouldn't interfere with each other, because they would be unambiguously started and terminated within a sound system where the end of every word can run smoothly against the start of any other word. If I were trying to make a stable sound system, the first thing that I would do is make the morphology self-segregating.

But if a self-segregating morphology is such a happy point, why hasn't any natural language come to that point? Well, that should be pretty obvious. No hack could transform a whole language into a having a self-segregating morphology. Or at least I don't know of such a hack. But even then, it's trivially easy to make one if you start from scratch! Don't you accept the idea that some things are easier to design than evolve (because perhaps the hacking process doesn't have an obvious way to be useful throughout every step to get to the specific endpoint)?

The exact mechanisms of phonetic change are still unclear, but a whole mountain of evidence indicates that it's an inevitable process.

That whole mountain of evidence concerns natural languages with irregular sound systems. A self-segregating morphology that flows super well would be a whole different animal.

Look at it this way: the fundamental question is whether your artificial language will use the capabilities of the human natural language hardware. If yes, then it will have to change to be compatible with this hardware, and will subsequently share all the essential properties of natural languages (which are by definition those that are compatible with this hardware, and whose subset happens to be spoken around the world). If not, then you'll get a formalism that must be handled by the general computational circuits in the human brain, which means that its use will be very slow, difficult, and error-prone for humans, just like with programming languages and math formulas.

Per my points above, I still don't see why using the capabilities of the natural language hardware would lead to it changing in all sorts of unpredictable ways, especially if it's not used for anything but trying to reproduce your thought in their head, and if it's not used by anybody but a specific group of people with a specific purpose in mind. I still imagine an engine well-built to drive its own evolution in a useful way, and avoid becoming an irregular mess.