Yes capitalism isn't about serving anyone rich or poor, or managing anything least of all the economy. Its about free markets - each person looks after their own financial interests and tries to make money and the big picture manages itself without needing central planners.
Yes, that is quite right.
The critical central planning, if you can call it that, is having a strong government which can create the framework of laws within which capitalism can operate. Draft the framework carefully, uphold the laws properly, have a small safety net for the weakest, and then let everyone then look after themselves.
Not to be confused with central planning of govt interference in the free market, which we all know is a disaster.