I assume you aren't familiar with how large studios handle development...?
The FFXIV dev team is split up into smaller teams dedicated to specific aspects of the game. The housing team is dedicated to housing (naturally), the gold saucer and other fun side content. They are almost certainly the team that developed this new mini-game. We know there is also a battle team dedicated to planning, building and balancing boss fights and their respective mechanics. There's a sound team headed by Soken. There are other teams too but we haven't exactly been given a detailed breakdown or anything.
As for which team would handle the pet glamours: I would imagine it'll be a collaboration between the UI team (if they have one, otherwise whoever usually handles UI) and whichever team typically makes new skills and animations. We don't know who exactly but it is certainly not the same team that made the mini-game.
The pet glamour system is not as simple an implementation as you seem to think. They need to design three new egi for SMNs with new animations for each skill (Yoshi has specifically stated this is why it's taking so long). They also need to build a special UI for managing the glamours and the mechanics for switching them. There will also inevitably be a quest to unlock the feature and likely a new SMN job quest to unlock the egi themselves. Even if they threw us a bone with some placeholder skills that do the same thing as existing pets (which they won't, it's a waste of resources) they would still need to make the new egi and create all their animations and sound effects. That is far from "a minute or two" of work.
Whichever team has been assigned to work on the system will have other priorities in 3.1 content for everyone. The pet glamour system will be sidelined until they have some breathing room from more pressing matters. Honestly I imagine it will most likely go into production once 3.1 is finished and will be scheduled for release in 3.2 at the earliest.