The "time it takes for the passengers of a generation ship to start a new civilization and gather the resources to launch another generation ship" is minuscule compared to the time to make the journey in the first place. Here's a Kurzgesagt video that essentially proposes using our solar system itself as our "generation ship" and getting from one star to the next would take on the order of a hundred thousand years, while the time to set up to travel that way seems reasonable to create in tens of thousands, or even thousands, of years.
https://www.youtube.com/watch?v=v3y8AIEX_dU