Like I said, put them into a ship with big enough storage to drop off a colony-forming ship every 10 generations - let them do the deceleration, mine your consumables, and re-supply the mothership. If that's happening every 10-20 generations, then you've got a release valve for your society (something that we don't have at the moment, but designing a society with release valves is one of the influences you can have across the millennia). And if (again, racing certainty) some of your would-be colonists get freaked by leaving the mothership behind, then the colonists have a release valve as they're establishing their society since there will be a re-supply mission accelerating back to the mothership next generation.
you've confirmed there's a hospitable panet (gravitational lens telescopes are your friend)
Short of manipulating a large (planetary mass?) lump of neutronium (which I'm not sure can exist), we don't have even a vague direction for such an object. And if we had to do that, we might well find it easier to go there (or send robots and relay stations) than to build such a telescope.
would you be happy if our lives today were bound to the vision of some ancient Roman emperor?
Some people seem to want to bind themselves to the pronouncements of some Roman carpenter, of whose existence we're by no means confident and whose diktats are based another half-millennium further back when (putative) his ancestors were slaves. At least we're pretty confident in the existence of the Roman emperors, even if some of them were as mad as a box of badgers. (I'm actually planning a walk along Hadrian's Wall - after that, I can securely attest to the existence of a Wall, with at least legion-marks referring to Hadrian ; after which, disbelieving in his existence would be perverse. In a generation ship, the existence of the ship, and it's constructors, would be hard to ignore.)
Like I said, that's why you build your society with (ir-)regular break points. Whether you have the ship travelling on a loop, or just driving straight(-ish) on for the horizon
Assuming 20-30 years per generation
Big assumption. The pressure to use medical developments and technologies to extend life is strong. On the assumption that the mammal body plan can't be pushed beyond 200 years, why would you go around doing momentous things like breeding before your 80s? Remember that for most of human history it was reasonably common to co-exist with your grandchildren, but seeing great-grandchildren was pretty rare. I'm trying to think of a mammal (or bird ; I don't know about reptiles or elasmobranchs at all, to cover the disparity of the vertebrates) that does routinely see it's great-grand offspring. If you wanted to change the generation ship people into a new species, that might be one of the most effective ways to do it - change life spans.