The point was that space colonies would be a solution to increasing lifespans which may result in an overpopulated planet. Space colonies is a way to accommodate everyone as biotechnology advances.
If there is ever going to be a solution to overpopulation, why limit it to Earth?
If it can't be limited to Earth, it isn't a solution.
'Overpopulation' is a hard thing to pin down; what size of population can be supported depends on too many factors for us to set a figure for the maximum possible human population. A more obvious issue is population growth - no matter what the limit of population size might be, continuous growth will eventually push us past this limit.
If population growth comes to an end (and, allowing for demographic lag, it has), then you can take 'relief of population pressure' off the list of reasons to go to space. There are plenty of good reasons left on the list - but population is not one of them in this scenario.
On the other hand, if population growth does NOT come to an end, then space isn't big enough to do anything more than delay the inevitable. No matter how much technology you imagine, there will came a point where the population hits a hard physical constraint - in the extreme analysis, you could fill the Solar System with concentric shells of Dyson Spheres, generating more and more room for humans to inhabit, but eventually the material required to keep building them would need to be brought to the solar system from so far away that it can't arrive fast enough even with speed-of-light transportation. If human population doubles every 30 years indefinitely, then a scant sixty years after we run out of space on Earth, you need three Earth-sized space habitats; sixty years after that, you need 16; and sixty years after that, 64.
It takes only a few centuries to reach the point where you need every scrap of material in the Solar System just for making human beings - and that leaves them nowhere to live, and nothing to eat; and then you need two Solar Systems thirty years after that; and four thirty years later again.
If population growth is a problem, going to space doesn't help. If population growth is not a problem, then it isn't necessary to solve it by going to space.
Space colonisation and population growth are unrelated issues - one is not a fix for the other.
Overpopulation worries a lot of people. That is because there is a sizable industry of population scaremongers who make a tidy living from it; the actual problem was solved in the 1960s, and the solution implemented over the last four or five decades.
It will take about another thirty years to complete the implementation of the solution, because of Demographic Lag; but the problem IS solved, and the data supporting this fact IS publicly available. Anyone who is currently worried by overpopulation is presumably enjoying being worried - because if they didn't, then they would look into the issue, and find out that their worries are three decades out of date.