“Private” isn’t describing the spacecraft per se, but the ownership of the spacecraft. “Privately-owned robotic spacecraft” is the “full” phrase, of which we elide part of the adjective.[0] Here, “privately-owned” is ORIGIN in that list, which puts it before TYPE.
[0] “Privately owned”, while on its own an adverb and verb, is functioning as a singular adjective in the larger phrase.
The spacecraft is privately owned, as the company is private, but I don't know that I would say that it is privately funded. NASA paid tens of millions of dollars, not just for delivering a payload (paying for a product/service), but for development of the lunar hopper, Grace, which was an award/grant.
[0] “Privately owned”, while on its own an adverb and verb, is functioning as a singular adjective in the larger phrase.
Edit: Previously said “funded”, not “owned”