Sorry, but there is simply to much historical evidence to refute this idea that population will naturally grow ad infinitum. The Myceneans, Hellenes, and Romans all had their periods of expansion and, subsequently, decline. The economic decline that occurred created spaces that, in your theory, should have been filled by new multitudes. They were not. Moreover, we can see that over several millennia that despite fluctuations up and down, the population over every state was more or less stable into the modern era. The population of India, for instance, was reliably between 125-150 million up to 1800. Only with the introduction of agricultural industry did population numbers take off. Moreover, we can see in many European countries that at the state of high capitalism, native populations begin to limit themselves. They must draw more from outside in order to maintain. There is no reason to dismiss the notion that in the Trek future, humans would exercise discipline over reproduction, keeping their numbers low.
Does it make sense to preserve population via immigration? Perhaps, but that would be ideologically driven.