Natural rights is an interesting topic, indeed. However, if there is ever a right that is entirely artificial, then it is property. The labor mixing argument has largerly fallen out of serious consideration in philosophical circles these days, being subject to way too many intrinsic problems.
Like, if I make a a glass of juice and pour it into the ocean, does that make the ocean mine, or did I just waste it?
Once you start looking at the enormous variety of property claims that exist, then you realize that property is better regarded as a cluster of disparate right claims rather than a single "has" or "has not" claim to owning something.
In context of the TFA, they apparently do not pause to consider that any property has to be grounded in some national law or another, or an entirely new body of law with their own courts has to be created for them.
Space has no need of such antiquated notions of property. Once we start getting there, we need agreements (and contracts) to delineate who can do what where, and those agreements will probably and hopefully cover vastly overlapping areas, not create artificial borders etched in stone like an Africa carved up by Europeans - whichi was and will be a blueprint for endless future conflict.