Shouldn't all Summoner's summons work just like Phoenix and Bahamut?
You summon them and they stay in the battlefield to auto-attack your target instead of going away. And then you would have to use another skill (which would be the equivalent to Enkindle/Rekindle) in order for them to use their AoE attacks. They only leave after 15 seconds and you have this much time to spend all your gems and your summon-related skills. This would make the job feel way more like a summoner than just a mage.
Also, Ifrit would be the one doing the gap closer, not you.