How many times do we have to repeat? It has to make sense in context of the current situation. My campaigns take people far and wide, to other planes of existence, sometimes alternate timelines. So instead of worrying about things like that or setting up false expectations that something will work when it makes no sense to me I replace it with something that does work on a regular basis and is far more useful.
We aren't watching a 2 hour movie, we're playing a long run campaign. I could have each of my 6 players having that one in a million chance a a few of times over the course of the campaign. The first time is luck, the second time is incredibly lucky the third, fourth and fifth time it happens to the group it's dumb as far as I'm concerned.
If it's something that's going to happen only incredibly rarely, then to quote you,
why do you care?
For that matter,
have you had players try to use these "one in a million chance" abilities four or five times in what is a presumably multi-year campaign? Or are you just making assumptions and nixing abilities based on nothing more than whiteroom theorycrafting? See, if people were pulling a "one in a million chance" four or five times in a 2-hour movie, that probably
would be very ridiculous (depends on the movie, though). But four or five times over
dozens of sessions? Nah. It's fine. Especially since it's not likely to even be four or five times.
it does not matter whether that is the only time the player wants to use the feature in the whole campaign, that does not suddenly make it more likely that they just so happen to stumble across someone they know just when they 'needed' it
Well, as Oofta says, why do you care? If it happens once in a blue moon, it's not at all hard to make it logical or just hand-wave the illogic away. Running into someone you happen to know is certainly no less illogical than anything else in D&D and it certainly becomes a lot more logical if you realize that NPCs can (and probably
should) move around just as much as PCs can. Travel for pleasure or business, being deported or transported, on the run from the law, being enslaved, magical mishaps... there's a zillion reasons why the PCs may run into someone they know that make complete sense.
You wanna know what's illogical? That fighters, no matter their origin, social class, and training, are equally proficient in all weapons and armor. That Eberron was the first official setting to take "magic is nearly 100% reliable" into consideration, even though magic has been nearly 100% reliable since the game was first created. That by RAW, PCs all knows how to read and write. That dragons haven't completely destroyed the ecosystem. That there are entire ecosystems underground that can support large, active life. That the vast majority of both official and homebrew systems assume "Medieval European" settings despite the vast array of intelligent beings with completely different histories and religions.
You know what the actual difference between these things and a background feature is? The difference is has nothing to do with logic. It's that easier to say that a player can't have an ability than it is to completely redo the proficiency system or to do deep-delve world building.