In general, I very much don't like games that tell me or limit me to doing just a particular thing. I very much like to have variety in my gameplay.
That said, a game should try to provide a minigame that supports each of the sorts of things likely to happen in its setting, and if the core gameplay doesn't support the conflict in a particular type of scenario or event well, then I prefer to have alternative rules for each scenario that is suited to the scenario, rather than trying to cover everything with a single set of overarching rules.
For example, 3e D&D has rules for turn based combat that work well enough as a tactical skirmish based minigame most of the time. The abstractions that they make as long as you remember they are abstractions are usually good enough for most events. But, they don't actually handle 3D combat such as flight particularly well, and so you have to have aerial combat as a special subsystem if it is going to come up often. Likewise, 3e D&D doesn't handle chases very well, because in a chase the problems with a turn based system are exaggerated to the point that it becomes a rather poor way to abstract things out. So it's good to have a separate system for handling chases and combat during a chase that uses slightly different rules that handle movement differently. And likewise, you wouldn't want to run a battle with thousands on each side using the same tactical rules you use in typical dungeon crawling, so you need a minigame for that.
While the use of optional minigames can get to be complex, I find it's less complex than having a single system literally trying to simulate reality, and more satisfying than having a system that is so abstract there is no relationship between the process of play and the thing being simulated.
But I am skeptical of the particular example, from Blades in the Dark. Having never played, I'm not sure how I feel about it, either as a GM or a player. Part of the attraction of playing an RPG for me is being in the moment and inhabiting the character. And if we pull a very literary device like a flashback into the narrative and disrupt the linearity of the story, I worry that would pull me out of the fiction and into the game in a way I wouldn't appreciate much. Imagining myself as a player playing the game, the mechanic comes as a bit of a shock, and I feel about it about like I feel about asking for hints in an Escape Room - I personally wouldn't do it. I want to pull off the heist using only tools that exist in "reality" and not tools that exist only in books.
The guy that taught me how to DM talked about his groups experience of B2 Keep on the Borderlands, and he said that his group turned the adventure into a heist game. He related this story to me in the mid-80's, and I can only assume the game itself took place say 5 years before when the module first came out. His group quickly decided that the Keep had better loot than the Dungeon, and spent the adventure planning how to successfully rob the bank. They certainly did not have specific heist mechanics for that, and they were doing it 40 years ago.
In terms of EXPLORING your character, these flashbacks are fine. Also, they're just a tool, and honestly I don't find myself using them a lot while playing my current character, who's function is basically just plain killing stuff straight up. Now, if I was playing a 'spider' I'd probably use those flashbacks a lot to show how he is a little super genius who thought of everything. Honestly, in the midst of all the back and forth about what the situation is, and who's doing what, and what the position and effect are, the flashback thing is nothing.
And honestly, I don't find there's a big dichotomy in 'generality' between RPGs. Yes, some present a pretty narrow milieu, and might focus on it to a degree, but the FitD engine, as an example, is not really like that. It can certainly handle any sort of situation that could come up in a game like D&D. As utilized in Blades it includes a fairly milieu-specific set of playbooks and crew types, but even the supplied ones are not exactly all that weird. You have Cutters (fighters), Leeches (techies/alchemists), Lurks (thieves), Spiders (Kind of a devious Sherlock type), the Whisper (A kind of skill monkey), and the Hound (kind of the BitD equivalent of the Ranger).
The crew types are equally archetypal, being Assassins, Bravos, Cult, Hawkers, Shadows, or Smugglers. I don't really need to describe those beyond the names, you already get what they are. Now, BitD is only, in its core rules at least, focusing on 'Scoundrel' PCs, but I don't see that as any different from 5e focusing on 'heroic' PCs.
I could easily imagine creating a Delving B/X style game using the FitD engine. You'd have various sorts of parties with different overall goals and orientation, and the PCs character types would probably be slightly more diverse, but you could create this game basically just by creating a crew sheet for the party, and maybe creating a couple variations of the 'Scoundrel' style character types. Its not exactly a radical rewrite.
Now suppose you wanted to use 5e to play something close to BitD? You'd need to recreate the crew mechanics, add rules for scoping out jobs, getting intel, interacting with the Bluecoats (cops), what happens when you have to 'do time', etc. You might also want to develop at least a couple subclasses and/or feats to allow PCs to specialize a bit more on their nefarious activities.
It is thus hard for me to see 5e as somehow being a much more general game in any sense than BitD is. There is certainly nothing inherent in the design of BitD which holds it back from working for a variety of genres. Sci-Fi, Supers, JAMES BOND!!!, some kind of X-Files type game, or various sorts of Urban Fantasy. These would all be pretty easy and wouldn't require any changes to the core rules at all.
There are in fact a lot of FitD-based games:
Court of Blades: courtly intrigue
Brinkwood: Rebels against the system
Mutants in the Night: Another game of rebellion, but you are a super
Misbehavin': Prohibition era gangs
Desks in the Dark: Magical High School
MortallyBankrupt: Some kind of offbeat existential horror about reality stars
Glow in the Dark: Mad Max style post apocalyptic survival
To Boldly Go: Star Trek
Runners in the Shadows: Cyberpunk game
The Final Frontier: Another Trek RPG
Harbingers of Twilight: Game of Magic Users in a Renaissance Era world
Laws of the Dark: RP in the world of Max Gladstone (sort of Urban Fantasy I guess)
Mothlands: Post-apocalyptic Bronze Age warriors
Sea of Dead Men: Pirates!
Copperhead County: Organized Crime in the modern US South
A Nocturne: Hard Sci-Fi Space Opera ala Andre Norton from what I can guess
Blades of the Inquisition: WH40K inquisitors
Blades of New Crobuzon: BitD set in China's fantasy city
Game of Darkness: Medieval game of guilds and intrigue
Blood Red Blossoms: You protect the people in a horror fantasy Japan
Replicant or Lesbian: Bladerunner-esque sci-fi
Streets of Passion: Adventures in a fantastical Los Angeles
A Fistful of Darkness: Weird West
The Messengers: Psychic operatives fighting a war in the modern world against ancient evil
The Typhoon Atolls: Sort of a fantasy RPG set in an Earthsea-like world
No Place Like Home: Horror/Mystery
Our Lonely Worlds: Outcast supernaturals against the system
Frontier Kingdoms: Demigod-avatar magical princesses balance tea parties and giant monster attacks to grow their kingdom.
Slugblasters: Teen hoverboarders exploring other dimensional gates
Swords Under the Sun: Heroic Fantasy
Blood and Sacrilege: Vampire RPing
Children of Midnight: Witches!
Studies in Darkness: I honestly don't know how to describe this...
This is just the list from the BitD web site, I'm sure it is not even close to a complete list.
Forged in the Dark has some more... It goes on and on, there are Supers games, Hard and soft Sci-Fi, Urban fantasy, Cyberpunk, Pirates, Wild West, Supernatural stuff of all kinds, Straight up dungeon delves, and on and on and on.
But this is what we've been saying: The structure of a game where the primary play loop is about the players engaging with what their characters are about, and the fiction building around that, is very inherently flexible! I almost guarantee you, with a modicum of dedicated mechanics you could turn practically any dramatic TV show online today into a FitD game (obviously some will be more compelling than others).
I suspect it is also a pretty easy system to 'drift' in terms of things like character survivability, power curve, etc. I would say maybe the most limiting thing about the FitD engine would be the fact that its dice pool mechanics only have fairly limited scalability. Since success is gated on the highest die out of N dice there's a pretty narrow range of dice pool sizes that work (from 1 to maybe 6 or so I'd say). To make a game that spanned a PC range similar to say, 4e, would probably require a bit deeper work, but I can think of some approaches that would work and not really alter the core mechanics of the system.