I think, conceptually, the idea is very innovative and different (which people seem to like these days).

Now hear me out guys. The idea does indeed take away from the original Final Fantasy sense of the summon, but it takes gameplay deeper in a different sense. It turns what is normally an individual activity into several layers of community. In a single player FF, it is understandable that you can summon these gods because you are the only one playing. (and I'm not trying to talk about lore or anything like that) Having something that a community needs to work together on to achieve, and also decide together when to lose what they have achieved I think is a good enough reason to sacrifice individual summoning of a larger scale. Not only can this have an impact on Player community reputation, it can improve a sense of a world for each server.

Right now, there are a few linkshells that are somewhat well known (a few I'm sure can come to your mind). They earn reputation through the things they have recorded themselves and shared with the rest of us. The majority of FFXIV players are not in the same server as those linkshells.
With these primals, I see the devs hoping that having something that can be spread by word of mouth can improve a given world's population and community. Instead of knowing linkshells from other servers and 1 or 2 from your own, people will know the free companies within their own servers because the word will spread about who has done what. Even a Free-Company made of only a few players can gain notoriety for catching a primal. My hopes are that this idea will help improve server pride and server politics.
Free Company rivalries, underdogs, and popularity can be a result of this. It's hard to explain how I see this abstract relationship between Server Community and Primals, but I can see where the devs are going.