You’re just not following the my metaphor. Perhaps it was a bad metaphor on my part. Let me try without it. This will be a bit long.
The economic system does not affect the government. The government effects the economic system.
Capitalism is an economic system. The government can then choose many different ways to approach capitalism.
It can be state-run authoritarian capitalism as is the case with China in their designated trade zones.
It can be totally laissez-faire, as in Randian fiction (or my favorite take on it, the book Futuristic Violence and Fancy Suits), allowing the market to dictate all growth
It generally is somewhere in between, with the state sometimes intervening in more or less extreme ways but generally just policing the system for health and safety reasons.
Capitalism, however, is not inherent to the government’s operations. A capitalist economy can change into a socialist economy, or a mercantile economy, and the government not change. The UK went from mercantilism to capitalism without a significant change in their governance, as a very notable example.