If you had FTL travel, you wouldn't use radio, but you could set up an FTL pony express. Or even something like a "conveyor belt" of microsats FTLing to a destination, dropping off their messages, and FTLing back with the replies. Whether that's practical or not and how fast it is depends on how exactly your FTL works, but it's hard to build a situation where you can FTL travel without having almost equivalently fast communication times.

Without FTL though, you're right, the communication situation gets real dicey real fast. Whether you're using generation ships or cryosleep or life extension to span the lightyears is irrelevant, by the time a colony ship reaches another system, both it and the humanity back on Earth will have diverged significantly due to lack of communication, and will continue to diverge as different events and pressures and developments affect each group differently. Imagine we started getting pen pals from the 1800s, but then to leave them a message, we have to wait for them to get to the 2020s to read it, and then by the time we get a reply maybe we'll be in the 2300s. What can we even try to meaningfully talk about? It's more like opening time capsules and studying history than anything resembling communication. It's interesting, but neither of our lives are at all relevant to each other. We are separated by too much time and without breaking light speed we can't break that.

Psychologically it is very difficult to imagine any two hypothetical civilizations not eventually separating and developing independently, because in all practical senses they will be completely independent of each other. And that's a manifestation of physical reality, there is really nothing we can do to avoid it. Not even an attempt at tyrannical oppression to make it otherwise can survive hundreds of years of delayed communication. Any "oppressors" sent will quickly end up having more in common with and more dependence on the colonists they are oppressing than they will with their "command" back in Earth-space. Even if they are successful, it's almost impossible to imagine they will not just end up as rulers of a separate independent colony. And then when Earth finds out hundreds of years later that their guys aren't quite following the orders they would've sent if they had been able to, what could they even do about it? Spend more hundreds of years sending more oppressors? Which will just end up doing the same thing? And why would they bother? If they want to loot their resources, is it still even going to be useful by the time they can get it back?

It's wild to think that after a few thousand years of independent development, we may not even recognize each other anymore, and we definitely won't think of each other as the "same" people. This will probably even be the case within the solar system, in some respects it already happens even here on Earth, nevermind once you start trying to stretch humanity across light years. Whoever those creatures that colonize other stars might be, it won't be long before we probably won't consider them "us", and they probably won't consider themselves "us" either.