No, the whole idea of getting into Alpha Omega Alpha is not to get into a competitive residency or to get some advantage but to promote scholarship and honor in medical studies. Participation in AOA activities is not limited to members of AOA and include lectures, fellowship and promotion of scholarship. In short, it's an honor and little more. It's not something that is going to "magically" transform you into an ultra-competitive residency candidate.
You don't need AOA to get into a competitive residency as plenty of people from schools that do not have chapters of AOA get into competitive residencies. Program directors of competitive residencies are not looking only at AOA membership and nothing else. You have to bring a complete package and AOA can be a part of that package but by no means is it all of that package. AOA doesn't prove that you will be a great physician or that you will be a top performer in a residency program. In short, the "something else" far outweighs membership in AOA in terms of getting into a competitive residency.
Most people in this country will not become members of AOA for one or more reasons not the least of which is that their school doesn't have a chapter. Having a chapter of AOA doesn't make the school any better (or worse) but is a matter of participation of the membership of this organization. It's not the end-all of residency and you can't just sit on your "duff" and smugly say, "I'm AOA and the residency programs are going to kill to get me". It just doesn't work that way. It has little bearing on whether or not you get into a fellowship (performance in residency is the key to fellowship).
Work ethic, performance, and interest/ability to perform well have more of an influence than AOA. Is it nice to have? Yes, but plenty of people do extremely well without it from any medical school in this country. AOA is no predictor of success in a competitive residency nor is it the only thing that can get you into a competitive residency.
In choosing schools, rather than looking for the presence or absence of a particular organization, particular board scores or anything else, look at where YOU can perform. If you hate your school and can't do well, you are the one that suffers and not the school. In the end, it's all on YOU and not your school or any organization that you belong to.
AOA may open some doors but those doors are not closed if you are not a member of AOA. You can open them with strong scholarship, strong work ethic and an ability to practice medicine at a high level. Is AOA helpful? Yes, but it's not crucial.