Well, think of it this way. Rod of Ages is just a plain, all around scaling item with no real special effects. Unlike utility through rylai's, abyssal, zhonyas, or the raw damage damage of rabadons, void staff, or ludens, Rod of Ages is a general purpose item. You can build it and see great success on about any AP champion. Name any mana AP champ, and Rod of Ages is a good buy. From TF, to Ori, to Syndra, to Taliyah, to Ekko. Anyone with AP can build it and be happy.
Why doesn't everyone? You build this item when there's really nothing better to buy and when you want to scale into mid game hard. It's not a great item for the gold you pay for it at first, and since this game is about snowballing right now, or rather has been for a while, it's not too optimal. But, build it on champions that usually take more than one spell rotation to kill someone (See Syndra for one rotation mages).
Anivia farms waves with spells constantly, so she needs the mana it offers. Syndra can cheaply clear waves with her Q, and Ahri with her Q. Ahri is also very mobile, so she doesn't need the health as much and Syndra deletes anyone so the health isn't so valuable on her either. But Anivia is slow and doesn't simply delete, plus the added tankiness synergizes with her egg.