Yeah, you should avoid needing that but it is certainly not impossible given high enough tech level & in-space infrastructure. Launch loops and orbital elevators should have more than enough capacity to enable that.
For a science-fiction-backed-by-real-science account of it might go, see this Orions Arm article: https://www.orionsarm.com/eg-article/49b46fd2198ed
Much bigger problem actually is where to put those billions once you have to move them in a hurry off a planet.
I think you're underestimating the effort, resources and logistics something like this would require.
In 2016, there were ~256 live births per minute[0]. That works out to ~369,000/day.
Even multiple space elevators couldn't hope to come anywhere close to that, but perhaps with launch loops as well we might be able to approach that.
But even if we could, how much of global production of, well everything, would need to be dedicated to getting even 350,000 people and the stuff they need off the earth every single day?
>Much bigger problem actually is where to put those billions once you have to move them in a hurry off a planet.
An excellent point. Even if you didn't need to hurry, the resources required just to stage such a migration are staggering and would likely exceed the production capabilities of the planet.
But as you point out, we don't really have any place to send all those "colonists" so the point is moot.
Does it? It's not an either/or proposition.
Researching and developing technologies to allow humans to live in space in a sustainable way seems like it would almost certainly also create applications of such technology that can benefit us on Earth.
Whether that's methods to scrub the atmosphere of a space-based habitat of toxins/pollutants, utilize waste in productive ways and a raft of other stuff would be incredibly useful on Earth.
As an aside, I'd point out that even if we start colonizing space habitats, other planets and/or even planets around other stars, there are no technologies available or imagined that could transport colonists from the Earth faster than birth rates would replace them and continue population growth.