good question. i actually agree with you. but free market is defined as "without government intervention", and some people defend that rigorously. i found this article where government involvement in a public parks fail: https://www.econlib.org/library/Enc/FreeMarketEnvironmentali... driving that point home.
personally i believe for every story in that article where the government failed or did worse than private owners we can find just as many stories where private owners failed to act in the public interest and only government was there to support the public. this is not a question of the system, but of the motivation of the people involved.
the article claims that government decision makers are seldom held accountable for broader social goals in the way that private owners are by liability rules and potential profits and i would counter that that is broadly not true in many places because people do expect the government to act in their interest and not doing so can and will backfire. even in a country like china where the government is more top down. but i digress.
so yeah, i agree that government and people are market participants. but i just had this argument in another thread where someone insists that i am a non-interested party in commercial property dealings if i am not a tenant or owner, while i kept arguing that as a person living in the city my interests are affected too.