Banks being public wouldn't change much. They would still have to be managed by someone, and that someone will have all the power that comes with money. At best this management will be independent from the government, and it would make no difference with private banks, at worst it would be controlled by the government, and it would not be banks pushing political power, but owning it.
As for preventing people from being rich and powerful, I'm sorry but you're living in a fantasy world. In all political/social system in history, from extreme communism to extreme capitalism and everything in the middle, there were figures of power, and power calls for wealth and wealth calls for power.
About countries having their own agenda in international trade, yes it's a reality. It's sad when it happens inside the european union because its point was to defend common interests, but otherwise it's part of the government tasks to defend the country's interests. And products exportation are part of it.