The problem is SE combined two classes that are difficult to balance a DoT class and a Pet class. DoT Classes always end up being mechanically broken or the developers implement them poorly while Dedicated Pet Classes end up being OP the draw back being it has a high skill cap.

The way to fix the Summoner is to make the Pets the focal point of the job while the DoTs become secondary just have Summoning Magic takeover. Give the Egis unique roles in the party.
Ifrit Egi- Single Target Bursts and increase physical damage
Titan Egi-Multi-Target Taking and party wide defense buffs
Garuda Egi-AoE Bursts and increase party wide spell speed
Leviathan Egi- High Sustained Single target DPS and Stuns
Ramuh Egi- AoE DoT Pet and Interuppts
Shiva Egi-Single target Tank inflicts heavy and slow