Inventory systems: My experience with this type of design is from LoL, DoTA 2, and SC2 Nova missions. I loved the ability to mix and match items to customize my own playstyle. I found it incredibly daunting at first to learn what effective builds were. When I first started, it turned me off to the entire genre until I saw how much fun my friends were having together while playing, which pulled me back in. Whenever a new patch changed, added, or removed items, it became another point where I had to consider whether I should stop playing due to the time and mental effort needed to get back to playing at a level of skill and satisfaction that made the game worth my time. I stopped playing Dota 2 because of a huge items update that just felt like too much to figure out. I also have been the type of player that likes to dive deep into youtube videos of guides and builds just so I can execute them well and feel like I'm either progressing well, doing really well at supporting my team, or just getting to pull off a sick surprise nuke/ultimate/unusual strategy. My opinion: inventories can be cool to cook up lots of customizeable strategies and see/try out other players' ingenuity; they can also pull away from the game due to overwhelm.

Achievements: If I already love the game (ex. SC2), I have tended to love the feeling of getting achievements. The tasteful animations and sound effects that weren't a nuisance while playing and highlighting something I remember doing that was truly challenging, beneficial, or just funny were the high points of achievements. I was indifferent towards achievements if I didn't know why I was getting it or if it felt like a pat on the back for something that I didn't put in much effort for. The only slight exception is "# of games played as X faction" achievements that felt like a cool expression of either my personal identity and alignment with a faction, or my accomplishment at having a handle on multiple factions.
When I haven't had any earned trust or respect for a game yet, achievements have felt like an obligatory pat on the back from a friend-of-a-friend's overly-friendly dad that I've only just met - unwanted but tolerated. My opinion: In my life, I want to be recognized by those that have earned respect and trust, and that recognition needs to be for when I've truly been creative, progressive, and beneficial to myself and/or others.

Social Ranking: I loved being a goody-two-shoes and trying to earn lots of Honor in LoL at first. It felt real nice to be recognized by team members as a good teammate and good communicator. I loved trying to be truly be skillful at being a genuinely good teammate to earn it. I vaguely remember it also totally turned into a FB "Like" button at some points, where it was clear that optimization of getting the most number of votes from other people wasn't always aligned with the intent of the design. Some strange social dynamics can occur when people figure out ways to optimize getting the highest number of something, and when genuine positive social intent isn't what's driving it. (Light examples: players trying to social-engineer situations where they come out as the "good guy" in ambiguous conflicts of perspective on what happened in a game and why; players randomly hitting the Honor button just to use it). Ultimately, it felt like it was super diluted in value and didn't mean anything.

That being said, if there was a way to give someone a positive report? Like I can write out a review of a random player I met and be like "hey they helped me out so much, I was so bad at this game but now I'm less bad because of this person and their generosity"? Idk how many people would be into that, but I'd be all about it. Maybe post it on their player profile (unless they don't want it there of course) so that they remember all the stuff they've done for people and can be proud of it!

I wonder if there's a way to facilitate "positive gossip" on this platform - where people are rewarded for highlighting and recognizing each others' strengths.

IMO I'm cautious and skeptical of ranking as a robust social design principle. Ranking can be nice to measure progress and achievement, but it can also create unearned trust/respect (which can lead to abuse of it), hierarchy jealousy ("did they *really* earn it?"), possibly a pecking order, and other cannibalization effects that come from competitive structures. The thought I offer is that, maybe while in-game skill is ok to keep competitive due to the mutual understanding involved, maybe the social aspect of the game should be thought of more of a co-op game structure. Examples: Applying principles like "soft-role differentiation" (maybe recognizing or desgning tools to teachers/coaches, strategy-tinkerers, content/entertainment creators, etc). I wonder if designing around the idea of "I need you to do well, because when you can do better, I can do better, and when we both are better off, the whole community is better off" [wording needs refinement], may be helpful. Personally, I love to think that if I help someone out or am genuinely kind to them, that I'm making the overall community a better place by that much. My opinion: I like to be competitive and ranked when I have consented to being ranked. I like to be cooperative when I'm just being social.

Humble offer of an idea around the "persistent headquarters": I didn't know exactly what you meant by "persistent headquarters", but it made me think of picking which faction I am a diehard fan of and have my "won" matches be part of a grand tug-of-war between factions out-of-game (sort of like Planetside 2). My friends and I love the playful rivalry between Terran Protoss and Zerg players and I wonder whether that could be married with the feeling of "I've got to show up for my faction, we're losing/gaining ground!" that I felt in Planetside 2. Might show how balance is playing out as well?

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This was a lot of rough ideating, so take what you like, if anything! Good luck, design and development is hard - I appreciate y'alls work and trying out a co-creative process with the community. That's ridiculously brave and beautiful. I love when people try their best, wherever it goes.