Well, the crux of the launch issue is time and economic expansion. You say staging would be prohibitively expensive, but that's only true at the time that you're building your single-stage mission. If we're building a rocket to another star we'd obviously be mining the solar system and expanding out into its vast supply of resources, creating a huge boom. Let's give that period a very conservative 5% economic growth rate. In 40 years, when the economy has expanded four-fold, launching a staged rocket with a mass ratio of four and the same payload will be the same relative cost as launching your original rocket, and be guaranteed to arrive much, much earlier. That earlier arrival time translates into economic and population growth at the other star, because there aren't any resources to exploit in-transit.
That leap-frogging of the missions just through economic expansion can be counted on to occur, even assuming no technological advances in propulsion. It's the same effect we see with Earth launches using chemical propulsion, which hasn't really advanced since the 1950's or 1960's. Even with the massive drawdown of launch funding after Apollo, the expansion of the economy means that we can keep putting mass into space at a higher and higher rate, even without concentrating on it, as long as there is a justification to expand into space. These same forces will be at work once we're living and working in space, at least by the time we can even contemplate an interstellar mission.
The implication of that is that the early, slow, single-stage ships will arrive at their destination so long after the ships launched later that their existence will be entirely irrelevant (10,000 new people will arrive at a star whose population is already in the tens of millions or billions), at least aside from the practical example their flight provides (we can do this! And oh, don't use terbillium coatings and remember to bring avacado seeds, because we just ate the last of a guacamole).
It may be that you're chosing the wrong destination. The nearest stars are guaranteed to get populated by later missions. Perhaps you need to switch destination stars to someplace obscure, or treat your mission as an ark whose destination doesn't so much matter as the fact of its existence, or design your mission to accomodate technological upgrades in-flight by including an ability to bootstrap new manufacturing abilities on board, so the latest Earth-tech propulsion innovations can be re-created with materials and equipment on hand.
You have to take into consideration that this slow starship will take 4100 years to reach Alpha Centauri, if what you say happens and it is bypassed by more advanced ships, the slow ship would still arrive 4100 years in the future around the years 6200 AD, by this time after 4100 years of technological advancement, I would be surprised to find flesh and blood humans populating the system, not that humans couldn't have got there, but if they got there 4000 years before the slow humans arrived then 4000 years of technological advancement may have advanced them beyond their physical flesh and blood human bodies, in other words they'd likely be no longer human, they might have uploaded their minds to machines or computers and become a society of AI, and as such they wouldn't need oxygen, wouldn't need an earth like environment, they could live anywhere that would allow their machines to operate, the humans arriving in the slow ship might find an already terraformed planet that was discarded by humans who had uploaded themselves into machines thousands of years ago, but these just arrived throwback humans could still use the planet.
Think of it not so much as a starship but as a one way time machine. The humans raised in the ship would be in a miniature world surrounded by 21st century technology, although they themselves have never lived in the 21st century, that would be the world they grew up in until they reached planet-fall.