I like the Summoner as it is really. The only issue I've really come across is Enkindle feels really lack luster for what it does, and Tri-disaster I wish was something more.

I think the Egi's do tell me who they're based off. Red flaming guy with Horns is clearly a spirit of Ifrit. The rock pebble man is a spirit of Titan, and the whirly bird creature reminds me of Garuda. Sure, they don't look just like them but they have the colour scheme and the theme down pat, I'm just summoning a shard of their essence if anything. If we get Levitahen-Egi I'm almost positive it'll be a floating snake with water whirling around it, Ramuh will be a purple man with a beard and a crystal staff, and Shiva will be a small, blue, dancing ice lady.

I'm fine with how the Egi's work as well. Titan is the solo pet, he's like the Voidwalker from World of Warcraft, it's fine to me. Ifrit is the Melee pet, and Garuda is the Caster pet. This I'd change in Heavensward. I'd keep them melee, and caster but change how they work with the Summoner. Ifrit should deal a bit more damage on his own and have slight cleaving, drop his Radiant Shield and give himself an extra steroid. Garuda should focus on making your DoTs deal damage longer like she does now and be the go to Summon for movement fights. You need burst damage, or you'll be cleaving little adds a lot? Use Ifrit. Prefer longer sustained damage, or melee isn't a safe place? Use Garuda. Then have Titan for solo'ing.

If we get a Heavensward summon, I'm sure it'll be Ramuh or Leviathen because of the story around Shiva. And they can be the summon we use for AoE. Then we'd have Ifrit for the Add Cleaving and being a "summon and forget" pet. Garuda would be the "Buff my DoTs to deal more damage" pet. Giving you extra cooldowns. Titan would be the Solo'ing pet. Leviathen or Ramuh would be the "Kill all these Adds" pet. Four pets, all filling seperate roles for different fights and playstyles. Obviously for the players who are progressing the hardest content, they'd probably swap pets depending on what they need. But people want their pets to be viable, and have uses, give them set roles.

I like the DoT aspect of the Summoner, we have the burst damage from the Black Mage already, they also have the elements handled as well. We got the other side of the Black Magic with the status effects and the debilitation. Build off that, give the summons a bit more purpose and make Enkindle actually good. I'd suggest Ifrit do a high potency attack on his Target, Garuda throw a DoT on her target that maybe makes them take a bit more damage from the summoner, Titan would be a defensive use maybe give the Summoner and Titan a sturdy shield for a bit, and then Ramuh/Levi deal AoE damage around their target. And make the damage from the offensive summons actually deal damage and have an impact.

So, TL;DR:

1. Keep the DoT aspect of the Summoner
2. Give the Egi's more purpose and feel different from eachother
3. Make Enkindle not suck
4. Keep the class as fun as it is (I really enjoy it compared to BLM)
5. Ramuh or Leviathen as our Heavensward Summon, and they should focus on AoE!