But then that just makes Summoner more of a support class that no one wants.

I dont know why this trend of Arcanist would be summoners class counter part, when it would most likely be evoker.What FF was Arcanist in and what did it do?

Evoker can have The command abilities attack, return, dismiss, Physical TP ability, Magic Tp Ability and 8 summons to pick from and still has 2 slots left, which can go to Elemental Siphon, or like you mention a Water Spell and Dark Spell.

Fire - Salamander (Final Fantasy Tactics)
Water - Bismarck (Final Fantasy 6)
Wind - Pandemona (Final Fantasy 8)
Lightning - Ixion (Final Fantasy 10,11,12) or Quetzalcoatl (Final Fantasy 8)
Earth - Midgardsormr (Final Fantasy 6,7,12)
Ice - N/A can't find a ICE based summon to replace Shiva. Maybe SE will make one?
Dark - Diabolos (Final Fantasy 6,8,11,12) Would be Fenrir but he was only Dark in 11. In other FFs hes Wind.
Light - Mist Dragon (Final Fanasy 6) I kinda wanna lead away from Carbuncle sorry guys.

And having pets out at all times would just be cooler. Cause to me, just calling them out to only attack then they leave is boring. Having them stay out makes me think its more like a partnership with the summon than just using it as a tool.
And god help SE if it only have 4 god damn summons lol. It better have 8 summons for every element. And Bahamut for a Non elemental! (Please Yoshi!)

They can design them (god help me ) like in 11 to have Physical, Magical, and Enfeebling attacks. They can cause huge mp to summon. Which can hurt cause of all the Heavy AoEing mobs like Ifrit, Garuda, CC, Chimera, etc.