Purpose of leveling up beginner job classes?

I know there has been a lot of discussion about how pulled jobs ultimately become stronger than any of the starting jobs. But is there any reason to invest skillseeds into the starting jobs or other "lower jobs". Does anything synergize or help other jobs within the same class?

From what I can tell, the only thing that carries over is weapons. For example, the apprentice mage learns different upgrades for sapling staff, which can then be used by other mage job cards, such as mage and black mage. Aside from weapons, is there any other reason to upgrade lower tier job cards?