I truly will never understand what it is about these airlines that brings out some super negative qualities in some posters... anyways
Alaska lost some of its luster after Delta hit Seattle. A fair number of AS loyalists switched over to Delta as it offered a better product. B6 has suffered some of the same from Delta. Both should aim at bettering Delta. Everyone knows AA could improve its product. Both Alaska and B6 do a few forays beyond their fortress areas, but really are not continental airlines. I can easily see all three improving their product particularly compared to Delta and providing superior services, AS western states, mexico and Hawaii, B6 eastern third of the US and Caribbean. AA everything else - but all doing a better job in their main areas.
What AA, B6, and AS are engaged in are codeshare agreements, the level of intentional cooperation you're suggesting would be illegal, but you're right that they do have complimenting strengths. I wouldn't really necessarily say that DL is a pain in the ass to B6 the way it is to AS. With JFK, both airlines have coexisted for a long time and B6 has its own niche. No one is really touching Mint which is why it has been successful but I'm not sure DL is even going after the same type of customer. B6 also has an extremely strong network to the Caribbean which has proven to be lucrative not just at JFK but at their other hubs. DL flies some of these routes as well (and so does UA) but B6 is the clear leader here. Additionally, NYC market is large enough for three large carriers to exist and if anything it has been AA that has suffered here at the hands of both DL and B6 and in BOS as well which is why I would venture to guess they've pursued this agreement with B6.
I don't think DL and B6 are enemies at JFK - for example, DL could have easily launched RNO, ONT, ABQ, etc. but they haven't and it wouldn't really make sense for them to do so. Considering DL's focus on business travelers and the presence of carriers focused on that same market with UA at EWR and AA at LGA, I don't think B6 is a huge roadblock for them and vice versa. B6 and DL product in coach (as B6 largely doesn't have business) are also not TOO different. In BOS obviously DL has been pushing B6's buttons but I'd be shocked if B6 management wasn't prepared for some network carrier at some point to make a push in BOS since it's a big business market with strong international demand that B6 obviously cannot meet. If anything I'd imagine they're glad it ended up being DL and not AA since AA already had a relatively strong base in BOS.
SEA, however, while a large market, isn't the size of NYC and DL and AS are going after the same type of customer. IMO the biggest roadblock for AS in California and particularly at SFO is the fact that DL is pushing so hard in SEA which forces AS to double down there. I think DL views AS as a much bigger target than B6 which I think would preempt any attempt from AS at SFO-ATL or LAX-ATL as another poster mentioned (along with their relative weakness in medium haul markets from California).