The Atlantis Expedition assumes that, barring a major miracle, they'll never reestablish contact with Earth, so they plan and act accordingly, by making sure they have what they need to become self-sufficient and raise and educate the next generation to some approximation of their standards, and making friends and allies and integrating some of them into their society. They plan it from the planning stages back on Earth, not from when they meet the Athosians or something like that, because that's a major attitude shift that they'd have to have from the beginning. So, prepared to educate and integrate, they take in the Athosians and make them an integral part of Atlantis, and random other people. Maybe they don't reestablish contact with Earth until (a lot) later. With each of the major cultures they accept, the culture of Atlantis changes, because it's not a "you're just staying in our city" thing, it's a "this is your city too" thing, and while there's a certain amount of picking up Lantean customs going on, the new citizens don't come into it with blank slates: the kids from M7G-677 aren't going to take "you're not old enough" as a reason why not, etc. Some people (Sumner, Bates) would have to be reconsidered, because you assign certain people if your goal is to be an outpost, and completely different people if your goal is to make an independent colony/culture. Sumner and Bates really didn't have the right attitude for the latter, although it's possible they might have had a different attitude if things were different.

And, because they have the means to teach and can always use trade goods and more hands to do the grunt work, Atlantis takes on students and teaches them (for a price, and the students have to work, and it's mostly science and practicals, because practical stuff is what they need, and science is ultimately what they need to defeat the Wraith...and those are the subjects Atlantis has the most people to teach), and becomes known as basically Pegasus's first major private school/university. Possibly the Genii are the first ones offered this, or send a lot of people to Atlantis. The Lanteans are a lot more cautious about the students, because they're not citizens, and although they might try very hard to only let the decent ones in, there's really no way of telling what agendas the students might have in mind. Consequences are dire for those who carry out plots against Atlantis, namely execution and blacklisting their culture from the University (and trade?).

Of course, the inevitable eventual contact with Earth would be...interesting, to say the least. Because, well, Earth sees them as a colony that hasn't been in contact. The Lanteans see Atlantis as home, and it's kind of weird to them that Earth keeps trying to exclude the non-Earthborn Lanteans.