Yeah, having played Blades in the Dark with a ton of 'navigating a ball/gala type event' and 'exploring the plot through verbal conversation' myself, it sounds like your main problem, @churchyardgrim , is that you're playing with people who aren't good at picking up TTRPG mechanics, and you may be one of those people yourself as well. "The game gets slow and plodding whenever mechanics come up during the stuff I'm interested in, regardless of what game we're playing" suggests that you or the people you're regularly playing with don't deal well with mechanics period.
I haven't played Monster of the Week, but if you have to think too hard about how to use Blades in the Dark mechanics in a way that serves your navigate-social-situation goals, then you may just not have played enough of it to be comfortable using its mechanics. There are essentially only four social skills (I don't count "Attune" but I do count "Survey," in addition to the obvious "Sway", "Consort", and "Command") and your playbook has a list of less than a dozen abilities, many of which are either downtime-specific or are merely passive permanent changes to your character sheet. The designers simply don't expect you to have all that much difficulty remembering what your options are.
My experience with social mechanics that slow interaction to a crawl is the social combat system from Exalted, in which attempting to wear down an NPC's Resolve to convince them to do what you want is nearly as complex and time-consuming as defeating them in martial combat. There are different NPC-specific strategy options for literally every NPC, including the random passers-by who didn't have a name until 2 second ago, and you have to spend actions figuring out what those are or possibly inventing new ones and then attaching them to that NPC. It's therefore impossible to internalize whether you're supposed to use [x skill] in [x situation] because every interaction starts with figuring out what this specific NPC's pushable buttons might be. It sounded interesting on paper, but I got overwhelmed quickly. Though even here, a big part of the reason it was such a slog was that I wasn't very familiar with the mechanics and had to keep stopping to look things up!
Despite the complexity of the fan-made cheat sheet below, though, note the instructions at the very top indicating that all of this is expected to take place within the natural flow of a conversation:
The point is that you are meant to put enough effort into learning the game that you don't have to slow everyone down looking things up every time mechanics become relevant. You'll need to consciously remember or look up a lot of stuff at first, sure, but it's supposed to become pretty instinctive, like looking over your shoulder to check your blind spot while merging lanes. Even crunchy combat is designed on the assumption that the people playing the game are going to learn the rules of the game eventually. You aren't expected to look up your own abilities every turn any more than you are expected to get out your Driver's Ed handbook to remember blind spot rules during every lane change.
If social mechanics consistently sour your experience of social maneuvering in TTRPGs, especially in systems as relatively 'crunch-free' as Blades, then probably what you want is a game oriented toward very freeform play, where the only game 'mechanic' is some kind of time-triggered prompting or external goal structure. You might prefer tabletop games that fall into what I would call the "collaborative worldbuilding" genre, like Microscope. These games don't really have anything in the way of social mechanics per se, but they have structural prompts and restrictions related to generating social interaction set pieces, to help ensure players know what social interaction within the game is for and what they as players need to achieve through the social interactions they play out.
I'd really like if my tablemates were more interested in those kind of games, but the players I have access to are more into game mechanics that affect play on a more granular level, e.g. "roll Diplomacy with a +1 status bonus from Glamorize and a +1 circumstance bonus from that successful Aid roll" (Pathfinder 2e) or "take 2 stress to push yourself on this Controlled Sway to change the effect from Limited to Standard, and you have a bonus die from your Trust In Me ability"(Blades). Consequently I don't have any recs in the "freeform social roleplay set into a structured prompt with an explicit game-imposed goal" genre that are directly related to gala infiltration, but here's what I do have:
Microscope, a game where there are rules about zooming in or zooming out to make either macro decisions about historical eras or micro decisions about what happened at pivotal moments in history. Players both make high-level worldbuilding decisions and also get to roleplay the pivotal moments. I never roleplayed infiltrating a gala but I did roleplay breaking into a pivotal galactic senate vote on android rights, and many other dramatic things.
I'm Sorry Did You Say Street Magic, a Microscope-inspired game where instead of a world history you are making a city and its neighborhoods, with structured points during which the players roleplay scenes set at a neighborhood landmark. I've only used it to design relatively modern cities with urban magic or near-future sci-fi vibes, but you could definitely do a more sword-and-sorcery city with dramatic factional politics and galas and such.
The Quiet Year and other Avery Alder mapmaking games like its sister game The Deep Forest. This one doesn't have much room for galas but I bet you could get some contentious community events going since the game is all about politicking in a small traumatised community attempting to rebuild. Conversations aren't freeform—you are quite limited in what a given opportunity to speak is allowed to cover—but you also aren't expected to remember any skills or calculate any specific bonuses. You just say what you think makes the most narrative sense within the game structure.
I don't actually know the proper genre name for the "worldbuilding"/"mapmaking" type of TTRPG. Also, though I like the worldbuilding element fine, I'm really mostly interested in the way it seems to create structure for roleplay without the need for much social mechanics to determine whether or not achieving the intended social goal actually succeeds. If others have suggestions for games that are focused on providing externally structured roleplay opportunities without offering much in the way of actual social interaction mechanics, I would love to hear more suggestions!