As your own article points out, and which should be correct, SpaceX has received very little subsidies (and Tesla isn't fairing all that well economically; so I don't know that there's much money to invest in SpaceX originating from there):
If Musk had a fortune that was <$5.5 billion greater than what it were, he would have had enough money to develop his Falcon rockets without government money in any shape or form. That's more or less the story with Jeff Bezo's Blue Origin (the company does appear to have at least received
smaller sums, but with Bezos' wealth, these certainly aren't needed).
I would say that it is correct that it is not simply 'the private market' that are driving the prices down, but very wealthy people like Musk and Bezos that have their own highly ambitious and futuristic visions (yes,
Bezos too). The cost of entry for this market is very high, so it makes sense that great pre-existing wealth is a requirement.
It is true that a very large share of launches globally (including for SpaceX) have state entities as their primary customer(s), but the relevance of that is, of course, heavily dependent on the precise argument being made.
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