I think you are missing something very basic about economics; you are assuming that resource consumption will universally go up as technology improves, when really that isn't the case at all. In fact economics is about making better usage of inputs to the point that there is actually more forested land today in developed countries than there was in the past, and not just because of NIMBY.
As the world economy continues to grow and technology continues to improve then more people will be able to have access to goods and services at cheaper prices with less impact. This is already true, were we to be using technology of the past we already would not be able to support the number of people on earth, and there exists already technology today that theoretically would allow the earth to support into the possibly trillions of people, at least in terms of feeding them basic nutrition.
With a developed space economy a lot of things are able to be moved off world that impact the ecosystem today. Mining and manufacturing, even some crop production depending on the transportation costs involved (a space elevator(s) (or if you are serious about stargates a stargate in orbit linked to one on the planet) would help with that) can be moved off world, as can energy production, among other things, all into orbit so that without any other worlds involved the standard of living on the planet can be greatly improved without further drains on the planet's resources itself; especially if there were shifts in social expectations and product design.
Having many planets at ones disposal allows for further specialization and hopefully gains from trade to be made, it also really allows for larger projects to be undertaken that are out of reach of a single planet with its labor force and resources.
So the average person may not be able to have their own private asteroid resort or something, but they might be able to afford going on a cruise that stops at various planets and see sights all over the empire. They could easily live a life that is in many ways better than that of the richest people today. As noted by the maps, fitting the people on the planet isn't a problem, and if the power and manufacturing is primarily off world and done by robotics people are going to be quite well off (assuming that is what the empire desires).