Unlike the US legacy big 3, AS does not have it's own fare filings on ex-EU routes (unlike US domestic airfares, most international airfares are asymmetric and differ based on your origin). It only has it's own fare filings to EU for trips that begin in US/Canada. So it can only offer other partner fares for trips that begin in Europe. It's also not part of the AA/BA revenue sharing TATL JV. Not sure if that is a factor why they don't offer their fares for purchase on it's website, but it may be part of it. Note it is has been observed that when AS offers partner fares for purchase on it's website, they are not always the cheapest fares available on that partner.