I think it depends what kind of flyer you are. Personally, as a US-based AAdvantage flyer, I always book them as AA*/BA (codeshare) flights because:

• If the trip cancels, I have more use with a credit that I can use on AA than one I can only use on BA
• My company prefers earning Business Extra points over BA's On Business (which are a pain to redeem--especially for upgrades)
• BA is bloody awful on OSOs before you get to the airport (they are at the airport as well, but at the airport being booked AA*/BA or BA prime makes zero difference). Calling AA (especially the EXP desk) is eons better than trying to deal with BA on the phone, which isn't even available 24-hours, and we don't have access to their elite desks
• BA is bloody awful with tickets purchased through a (corporate) travel agency. They're trained to say "NO" to anything. Working with AA is far easier
• If you're using AAdvantage miles to upgrade a BA flight, from what I understand having it coded as AA*/BA should make it easier
• BA op-upgrades...what op-upgrades to non-BAEC members?

I fly in J so the lack of for-sale upgrade offers was never noticed until I read about this in this thread.