There is room for a pet job like this. I just don't think it should be the Summoner. Summons in older FF games would show up, use their cinematic attack, then go. It felt like you were calling in a favour from an ultra powerful god like entity, there was never any kind of companionship or interaction between you and them outside of them turning your enemies into mushroom clouds.
I think the Summoner should be that in FF14. You point and the heavens split and Ramuh smites your enemies with a lightning bolt.
You hit the nail on the head with the reason - Any pet job should have their pet as an actual companion. One you can choose, level up, customise their skills and how they engage in combat and personalise to your hearts content. Similar, perhaps, to a BM hunter from WoW, where you can pick and choose what you want as your pet, what it looks like and which skills it has to end up with something unique to you.
There's lots of unique interactions they could add to a job like this. Dashes where you get a quick ride from your pet, coordinated double team attacks and positional bonuses depending on where each of you are stood. Out of combat flavour skills like feeding it, grooming it, perhaps even emotes and poses for your pet too. There's a lot of room in this space for a functional job in terms of it's abilites, job fantasy and asthetics.
The Summoner wasn't that kind of pet class. It never interacted with its pet outside of simply pointing. And that's fine, I want them to lean into that fantasy and that design more. It opens up a lot of design space for a more traditional pet job, as well as DoT orentated Mages and allows them to have that job idenity all to themselves without Summoners stepping on their toes.