I think the difference between the “private” development of Dragon and the “government” development of Mercury is exaggerated.
Like Dragon, the Mercury capsule was (mostly) designed and (entirely) built by a private contractor, McDonnell Aircraft.
Like Mercury, Dragon was designed and built with input and strict oversight from NASA, including a contingent permanently on-site with the contractor. NASA always had total visibility and the final say on everything.
There are some important differences in the contracting models of the two programs, but a lot of the “private”/“commercial” framing of the ISS crew/cargo programs is just leftover vibes from the 2000s when ”harnessing the dynamic private sector” was the way to frame a new program that you wanted to get funded.