I distinctly remember playing FF10... and there were no pets for Summoner not in the classic sense.

Did play FF11... in that game the SMN was a half assed White Mage with no abilities. And the Summons were pretty weak and drained your MP too fast to be used as actual Pets.

Never played 12 or 13...

But I have played pretty much all of them previously.... All of the Summons were little more than large scale nukes with a one shot AoE attack... and I linked them for you...

I do not know here you can sit there with a straight face and somehow claim after saying that it doesn't "feel" like a Summoner.

The one that doesn't "feel" like a Summoner is a half assed pet class from FFXI.