The issue, in my mind, isn't necessarily, "who does what when". We've all been around long enough to know that procedures are, to a certain extend, arbitrary. In other words, DAL, UAL, AWA, etc, all operate the 737, and they all operate it a little bit differently. One manner isn't necessarily better or worse than another. The important thing, is that within a company, within interchangeable crews, everybody knows what to expect from their fellow crew members. That's important.
If AA feels like the Captain should be the voice of the airline and make all the PAs, fine. If AA decides that the Captain will call for all the checklists while the airplane is on the ground, great. If AA deems it appropriate for the FO to do all the walkarounds, that's their right.
However, the upshot is that the FO is denied any opportunity to "manage the operation". While I'm assuming we can all fly just fine, the need to "manage the operation" is critical for airline crews and critical for FOs upgrading to Captain. There is no reason why many tasks cannot be delegated based on PF/PNF basis versus CA/FO basis. The Captain is still the Captain and has final veto power over any operational decision.
Personally, I think if the FO is truly "a captain in training", then WHENEVER POSSIBLE tasks should be assigned on a PF/PNF basis.