I could easily come up with multiple answers for each of these, but I'll pick the biggest one from each category.

Change: Diadem. It has a ton of potential, but the implementation is sub-par.

1. Make the objectives be the driving force behind player activity. Award chests/gear/spoils for the completion of objectives. Let the list repopulate when you clear some or all of them.
2. Scatter around a decent number of randomly occurring events, similar to goal-oriented FATEs (rather than "kill a bunch of stuff" type FATEs), some of which include gatherers in their objectives. Make these also award chests/gear/spoils.
3. Give battle classes and gathering classes more reasons to interact, in addition to the events above.

Remove: Specialist-specific recipes. If you have to restrict access to something to force people to use a system they'd otherwise ignore, that suggests that there's a problem with the system, and your efforts should probably go into making the system interesting and worth using on its own merits, rather than handcuffing it to something else.

Add: Some sort of alliance-based, tiered, popped, claimed-by-the-group-that-popped-it NM system. Something along the lines of the ZNM system from FFXI, minus the ridiculous camera mini-game nonsense that was necessary to purchase the pop items.