yeah but a single ship as primivite as we can make today, if even deep-space worthy at all, won't be able to change it's direction or improve its detection technology along the way with the resources avaiable.
ofc there is also the fact that if the world were not to end in 12 years, even a ship build 100 years from now would be able to overtake a ship we could build today easily. thats the really scary thing about generation ships, that when you arrive you find a fully developed civilization of your own species that treats you as a archeological curiosity.