Yeah, private detective is an open compound:
https://en.wikipedia.org/wiki/Compound_(linguistics)
At least I think that's the right term (I hunted around for one that fits). So, is private spacecraft also a compound? Is it idiomatic? Maybe. Another example is little black dress, where "my new little black dress" sounds right and "my little new black dress" seems to refer to a different kind of garment.
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.