For me, I felt with SCH and SMN having magical pets (even though Ifrit and Titan are melee-ranged) and MCH having physical-ranged turrets, I wanted a physical, melee pet to differentiate it. I also wanted to make a pet job were both worked together and not just a master and turret and a pet job where the master would have far more control over their pet for a majority of the time, versus SMN or MCH who either place-and-forget or just control it for one ability (Garuda).
The problem with using the three examples you used, Luca, Cait Sith and Reeve, and Lulu, are that none of them actually used said puppets or dolls in a way that could be made into a mechanic.
Luca only every used Calca and Brina (who were both separate partymembers) in bands (group attacks) and specialized in doing physical damage on her own. I had originally thought of having a Summon Calcabrina skill for the level 70 ability but thought it would kind of steal from SMN (and I thought we don't need another big pet blocking partymember's view).
Lulu was a Blackmage and only ever used her dolls for standard attacks and channeling for her spells. I feel that her dolls were cosmetic and weren't an integral part of her gameplay.
I feel the same way about Cait Sith and Reeve's relationship. As a partymember, Cait Sith was a standard squish mage with luck-based Limit Breaks. However, I did use Cait Sith's Slots limit break as inspiration for the ability Elemental Slots.
With the look of the Puppet, I envisioned that any kind of puppet, doll, or automaton could be used, with the look of the user's Puppet coming from what main-hand weapon they have equipped. I imagine things like some of Lulu's Dolls, XI's Automatons, and XIV's Mammets as possible Puppets the player can use (maybe even some of XIV's Minions too). With this, there wouldn't be a demand for pet glamour either, with people just having to glamour over their weapon.