> wouldn't take one or two hundred years to become fully independent for the necessities to survive on its own

My concern is the incentives and economics that make it possible for them to exist, principally (relatively) cheap launch and transit costs, preclude it ever being economical to invest the resources required to ship say modern chip lithography equipment out to the new colonies rather than them continuing to buy them from Earth. Or the alternative, they get treated like pre-revolution US colonies and are intentionally kept dependent by restrictions on imports and tech exports in order to provide a market for the mother civ to sell into. And this time instead of being in a verdant land full of resources the colonies are on baren rocks dependent on the mother country for critical life support supplies so the capture and dependency are magnified and much stronger.