Well, we do have a Pet Week, but yeah… it’s barely visible except the bonus experience itself.
From the only tidbit right now it seems they want engagement. But it’s like… everything requires it in one form or another.
What have been the most recent features and how did they gain traction?
- Hero Talents: Direct character progress, unavoidable.
- Warband Features: A lot of quality of life improvements with a little bit of gold sink. (This will be Transmog in Midnight)
- Delves: A separate tier in The Great Vault (replacing PvP) and hefty rewards at the highest forms (crests, keystone, titles, toy, mount skin), seasonal limited time goals like in the other modes, but you’re not confined to the current season instances
- Returning Plunderstorm & Remix: Limited time and a different approach to character progress. Alleviating the randomness of sought-after drops in the case of Remix.
- Timewalking: Not much new within the system itself other than Classic and Battle for Azeroth entries, but they also added new rewards to the previous expansion vendors. And then there’s me still without the crispy drake.

Of course any ‘walkable’ content can also be a Trading Post goal, but that’s hardly a deciding factor.
In short, their most precious tools of choice are 1) periodicity 2) a collection of diverse but loosely connected rewards that while they tell you they’re a part of the same micro-universe, they can stand on their own (Usually. We don’t talk about that one drake recolour.).
This is not impossible. Well, to apply limited time to pets you go out and battle (as opposed to drops) you’d need something like Globe Yeti, “time of the day” spawns, Legion tamers, etc. This is probably the least annoying way to limit the content (maybe except the latest, don’t make it random!) to be honest. A set of 7 or 14 pets each appearing in a specific day of the week doesn’t seem particularly out of the way for either side of the fence, though it may be suspect to time zone woes. We’ve already had a taste of this with the Wanderer’s Festival.
As for the later part, though, well. I can think of a way. My way would definitely span a lot and be of roughly Darkmoon Faire’s level of scope. Or maybe twice as that. I don’t feel this is the time for this, right now. But I think the current system’s mistake is the concept is too rigid. Train this, conquer that. A couple of toys and a tabard, mostly related to the thing itself. No place for taking a step outside and thinking: “When can I use this elsewhere?”. The Delves currently have the same problem, but at least the mounts (and the carrot of help with gearing) are fairly decent. Ditto Plunderstorm, but that’s by design and the limited time is the more important “motivation”.
I don’t claim we need to throw crests into your dungeon bag. And I definitely don’t claim we need something as flashy as a mount. It didn’t end up well last time. But they’ve lately been churning out endless transmog sets which have different colour variants scattered around Trading Post, world events, promotions. Enticing quite a few to ‘complete the set’. It’s dirty, it goes against the entire “internal continuity” thing but it works. I am sure they could throw in a couple of house decor, and they said the old accomplishments will get us some… It would be a nice cameo if the pet portals that used to expand your stable, now came back as portals that your pets may pop out of, OK, now I am just being silly. 
Still, though, it seems that unfortunately the biggest barriers are related to the interface. Being told to install a bunch of addons (everyone’s fav topic right now, har, har). The bizarre interplay of Base Stats and Breeds, whichever is more important depending on the pet. No team sets. The minimap LOVING to disable “Track Pets” (a way to create a self-fulfilling prophecy). The long animations, sometimes bugging out or skipping numbers (one of the things nobody’s figured out yet). Tooltips lacking essential information (Apocalypse or Haunt come to mind, but also you need to “trust me” when I say the base crit chance is 5%). And I could see this being the problem. Because that sounds like the least fun thing to work upon (unless you’re Mutanis or Lazey :P).
I am sorry, I started with talking about the weekly, so I should tell a word or more about that. Other than including PvE elements, similar to the Timewalking weeklies including raid bosses, and making the reward more akin to the other ones, I don’t have any other ideas. And I’ve just noticed, unless Wowhead is iffy, the two mainline PvP weeklies are all limited to PvP rewards, too. That’s a little sad. Maybe just throw extra Resonance Crystals (or whatever the Midnight form is) to all of them, I am not sure. Or (applied to everything) weekly event-exclusive currency similar to Tokens of Merit, but spent on Trader’s Tender, heirloom upgrades, and unique collecting-related knickknacks such as resetting the cooldown of a chosen toy.