We’re never going to leave this planet/solar system if we don’t discover FTL (Faster Than Light) travel. Pretty scary if you think about how ridiculously empty is space.
If ~1.0C is the fastest man can travel, that's still pretty good. Alpha Centauri is in reach (less than five light-years).
Maybe it would make sense if you could convert the mass of Proxima Centauri into rocket fuel to fund more expeditions? That seems like a fairly long term plan though.
Exploration is what humans do. If Alpha Centauri is only 5 years away, you can bet we're sending people there no matter what. Permanent colonization only becomes a priority in an existential event, and if we eventually achieve ~1.0C travel then that implies our tech is at a level we'd be colonizing Mars and a few moons long before we think about leaving the solar system.
Even getting to 0.9C is going to be very very hard for humanity.