Apologies in advance for my massive text. TL;DR at the bottom.
Let me try approaching from a more technical aspect why a "Pet melee job" in most cases is usually detrimental to it's own balance and by proxy results in its over all weakness.
When speaking of most other Melee DD classes, their power is defined by several aspects which can be controlled to some degree (and others that can't). For example. When warrior is introduced as the Primary Melee Damage power house, it'll have abilities and inherent traits (Stats or passive traits) that make it naturally fit into this role. From there a player can then use Gear as an enhancement to further improve it's effectiveness.
When other variables come into play like Mob level difference or certain special moves or mob gimmicks, the player through skill and other means can work around it without compromising too much on their over all power.
Using XI as an example (Only because it seems to be the one example everyone is familiar with) When you encountered an HNM, all DD's had a significant damage loss than if they were fighting an Even match. This was expected, as some mobs were designed with higher defense, on top of their already steep level difference. But it affected all DD's the same. From there the DD's abilities, gear, and food allowed it to fight this "gimp" as best as possible.
When making a melee pet class there are several things to keep in mind to maintain balance. The first being that "when pet and player damage are added together, they are equivalent in balance to another melee damage class". So in some way, the Player has to be less potent than other melee's by itself to justify the need for the pet. On paper this doesn't sound bad but other variables later make this problematic.
The other being that augmentation of the pet (and player) has to be balanced between other classes as to not allow that either the pet, or player (or both) far exceed the damage dealing capabilities of another Primary DD class. In XI, Pet augmentation equipment was rare until recently. You'd hardly find any equipment that said "PET: Accuracy +" or anything that would help improve it's DD for this reason. This often came at a cost as well. If you used a specific piece of equipment to enhance your wyvern, you were losing a vital piece of gear for yourself. If you made a piece that gave a decent amount to both, you run the risk of the combo of player and pet becoming too powerful than any other class.
This becomes even more compounded when you factor in that a pet can't gain Food advantages like a player.
When a Melee pet class is presented with a situation of HNM's where level difference penalties and over all high stats of a mob cut damage, a melee pet DD recieves this penalty twice. One for the actual player, and one for the pet. As mentioned before, the player can compensate with food and better gear. But for balance reasons this poor pet has almost nothing to help it improve and compete. Even if the Player compensates as much as possible, the design of the combo relies on "both" dealing damage together and as a whole the combo is far less powerful than any other DD that just has to worry about himself.
Then worrying about handling things like Stray AoE's or status effects affecting the mob as well. If the pet dies, that damage is completely lost until it's called back. Healing the pet becomes inefficient because you have more MP being spent (assuming direct cure is possible) to keep this mediocre damage dealer alive just to justify having the Melee pet combo there in the first place. Over all MP inefficiency is detrimental to EVERYONE participating.
All in all, The variables and various components it would take to balance the class and actually make it worthwhile to play are so numerous that it becomes nearly impossible. Having the pet comprises your job, not enhances it. If you want your class to actually be balanced and liked as equally as any other melee Damage dealer out there, You should understand that having a pet will only take that away.
TL;DR - Having a pet with a melee job just causes too many problems and there are too many variables to appropriately account for game balance. In the long Run, if Dragoon does get a wyvern pet, it will ultimately bring the class down as a whole.