I agree with the OP.
For those of us who have played most FF games, this class feels nothing like what a summoner is supposed to be. It completely betrays the theme I think. It's underwhelming.
There are small things they could do to fix this I think:
1. Increase eggi detail and size.
2. Allow Enkindle animation to transform eggi to a full primal for a second. Just this animation change alone would make a big difference in the feel.
More complex changes:
1. Shift more summoner damage away from the player and more toward the eggi.
2. Allow for summoner to teach the eggi skills (similar to FFX), and make it more of a class whose number one learning curve priority is learning to customize and control your summon.