Summoner in FFXIV makes a lot of sense if you think of it as a specialization of arcanist -- not its own standalone entity. Plenty of MMOs have base classes with different talent trees or specializations, where you get to pick and choose what type of role you want to play. FFXIV is different in that there is no wiggle room; once you choose your specialization, that's it. You equip the soulstone and you are a summoner. The summoner job in this game makes a lot of sense if you take the summoner quest lore, general world lore, and arcanist lore into account. As others have pointed out, summoners across final fantasies have all varied greatly. FFX in particular introduced the idea of a summoner who actually kept a pet out, protecting the party. FFXI solidified summoner as a 'pet' class.

What's really interesting is that OP wants summoner to be MORE like the Final Fantasies of old, while the usual critiques I see on the class is that not ENOUGH emphasis is given on the pet as a separate entity. As a level 50 SMN that pulls off 300~ dps in any given fight, my pet does about 50-60 dps. That's less than one third of me, but... raw numbers don't take into account all of the added bonuses your pet gives. Extra burst damage from Rouse/Spur/Enkindle and extended dots from Contagion are worth more than players may at first realize. Dying and not having your pet out with you right away once you are raised really hurts, and it does a noticeable blow to the summoner's effectiveness.

I am happy with the summoner pet class in this game. Although it is very reminiscent of the functionality of warlock in WoW (dot pet class), it does have its own flavor that I quite enjoy. It is unlike any other class, and drastically different from the only other caster. The lore fits the class and while everyone is entitled to their opinion of what summoner 'should' be, SE has proved over the years and many Final Fantasies that there is no 'one' right answer. My personal favorite type of summoner is one that always has the pet out, so I suppose that is another reason I like how it was represented in this game.