It is true that immortality is not actually possible to achieve, but it's certainly possible as a direction. Maybe Breeding vs. Immortality would be better names.
War is not left out; for the purposes of this conversation (since we are standing outside of culture at the Archimedian point of arbitrary leverage...) we can consider it one of the options available to those who wish to have more and shorter-lived individuals.
Malthusian catastrophes have probably overtaken plenty of populations -- Jared Diamond makes good arguments in Collapse for, for instance, Easter Island, the Mayans, Greenland, etc., having fallen victims to nonsustainable growth in constrained areas. That there has never been a global modern Malthusian collapse does not mean there won't be one at some point -- arguably we have avoided one thus far by "a continuing series of miracles", as the phrase goes about Israel. We have had several uninterrupted centuries of globalization and technological advance during which there has been lots of low-hanging fruit. Malthus, too, "only has to win once."
As for why, Levi, we limit ourselves to a "stable" society, the ultimate reason is in the title -- the second law of thermodynamics. Even assuming Golden Age SF miracles of FTL, utopian social orgnization, etc., eventually entropy gets you. There are lots of reasons, however, to presume the border is a lot nearer than that.
Assuming no FTL, the limit case is that we convert our living space into a sphere of pure optimized computronium, or whatever, whose radius is expanding at lightspeed. Malthus's spectre still applies in that scenario: the available living space is still expanding geometrically, while the potential for new individuals is exponential. At any given moment there is a 2/3-pi-r-cubed volume of intelligence within the sphere competing to breed into the new lands emerging at the 4-pi-r-squared surface area; we have a cube-square-law problem.
In the near term, of course, the picture is a good deal less optimistic; we are in a highly livable little oasis surrounded by a vast wasteland unbelievably inimical to life. Manned space flight is in fact no kind of answer at all to population growth, and in the near term it is also no kind of security for "survival of the species". The gap between going to the moon as tourists, carrying our air and food with us, versus surviving there an apocalypse that would overtake the earth, is like the difference between watching the Olympics on TV and winning a gold medal yourself.
10-11 billion peak population followed by slow decline is fine, but is assumes a very modest increase in potential life expectancy. I think classic SF, for cultural and narrative-technical reasons, underestimates the plasticity of humanity as much as it overestimates the practicality of space settlement. Star Trek's vision of people with hundred-year lifespans jetting about the galaxy with free energy is wrong in both directions. I expect we are likely to get potentially thousand-year-lifespans long before we can produce a viable wholly independent offworld settlement.