You are correct in surmising that Free market capitalism is completely independent of of government. Marxists tend to confuse this because of the unholy welding of an economic system into a form of government that is Socialism. Free market economies when they have existed have brought about great improvements in standards of living, knowledge, innovation, science, medicine and tended to bring about a much happier populace. These economies have existed under many forms of government but there are three critical elements to make a free market work. Capitalism is not itself a free market. A free market capitalist economic system is just one of several forms of capitalism.
To establish a free market these conditions must exist and be maintained.
1. Control of monopolies to prevent collusion, unfair trade practices and intervention.
2. Limited government involvement in the economy.
3. Low levels of corruption.
4. Sufficient capital in the hands of upper middle class and wealthy to create new businesses and invest in existing businesses.
5. Lessened or total lack of class system restraints or some sort of loop holes that allow common people to rise up as ability and luck allow.
6. Sufficient trade partners or an economically sustainable internal customer base.
Socialist Democracies cannot provide any of these conditions. Having a strong central government which is necessary for a Socialist anything means there will be high levels of corruption which leads to Corporatism. This kills the free market aspect. Capital is sucked up into taxation leaving only the very wealthy able to invest and sustain business. Socialism creates class systems and becomes a barrier to upward or downward mobility of it's people. Staticism is the very goal of Socialism so by design it is meant to keep people in their place. The lack of internal capital makes it very difficult for an internal market and only by temporary measures can a Socialist nation like China create favorable trade conditions. The gross inefficiency however of the government will eventually kill these conditions.
As such a Socialist nation is incapable of sustaining a free market economic system. The very reason man of our protections are so closely guarded in the US is to prevent the very things Corporatism and Socialism bring to a nation. A free press is supposed to guard against corruption but in the US today the media is OWNED by the very groups which control our government and influence our political parties. The use of blackmail advertising has created a situation where the press has repeatedly been silenced by corporate and special interests so as to abet corruption and wrong doing.
I'll disagree with you on climate change. It's not carbon it's deforestation and the so called efforts to battle climate change do far more harm to the environment than the problems they are meant to solve. Most of it like carbon credits is nothing but a giant get rich scheme and political means to punish groups and industries which support right wing groups and agendas. Punitive actions against the "wealthy" by the wealthy for the wealthy. It is a battle for dollars not the environment. I am a former member of many environmental groups but left in disgust over the politicization of the movement and the way it has become nothing but an arm of the DNC. The environment is far too important to make a political play toy. The climate change movement has done exactly that. Made a mockery of every principle for which environmentalism was supposed to stand for.
Unions served a vital role in the past, today however many are nothing but an arm of the DNC, thus the opposition and union busting. Unions are one of the biggest contributors to the DNC and the idea that if you want a job in a union shop you have to know somebody there goes against every principle America was founded on. Compulsory membership in unions defeats the very purpose they were formed for and has replaced the robber barons with the union barons. If you are part of a union it means you actually work in the trade you represent. Most union leadership never worked a day in their life and have no clue what the people they supposedly represent want, need or think. They nothing short of a parasitic corporation with all the nasty side effects of big biz but none of the redeeming qualities.
Economic systems should not and cannot effectively govern. A huge fallacy of both parties is the attempt to allow business to govern people.