I'll grant that it definitely sounds ambiguous, but I actually think the phrasing "robotic private spacecraft" is more correct in the end.

I think this is a fair analogy: suppose we were talking about a "private detective". If we were writing a sci-fi book, we might talk about a "robotic private detective", but "private robotic detective" would sound odd.

Now, I'll grant that "private detective" has a lot more cultural weight than "private spacecraft", but I think it's fair to say that at least the word "private" is playing a nearly identical role in both phrases. With that in mind, I think "robotic private spacecraft" makes sense.

I suppose you could take this argument one step further and resolve the ambiguity by asking which distinction (robotic/non-robotic, private/public) the article writer thinks is more notable and placing that first.