The numbers express the general attributes and abilities of a character. There is no difference between a 13 and a 14 from a character description standpoint, although there is obviously a numerical impact on game mechanics.
Which does affect how it models a concept. If you want a 'strong' character, 13 might let you lift a bit more than 12, but the +1 bonus is probably more significant.
But, it's also all relative. You are a 'strong character' if you have a 14, and the rest of the party has 8,10,13, and 16. And the guy with the 16 is clearly also a strong character, and can feel safe in his concept of 'big dumb guy.'

OTOH, it's harder to paint yourself as a 'strong character' with your 14, when the next-lowest STR in the party is a 16, and everyone else is 18+. You're prettymuch the weak guy, at that point.
Of course, it also gets into the world, if you're constantly meeting 18 STR blacksmiths and 24 STR town guards, it's different than if the strongest NPC you ever meet has a 14.
But there is a difference between 'above average' 11 and 'strongest man in the village' 18.
In a village implied by ordinary people statted with the ol' 3-18 bell curve, the strongest
man probably doesn't have an 18, unless it's rather large for a medieval village. Maybe a village of 500 or so, including women and children.
My concept is 'the strongest man in the village'. The village's population is around....let's see...two hundred-and-sixteen. What number models that?
Since you specified /man/ and villages tend to have women & children, too, maybe a 16 or 17. If everyone's rolling 3d6.
OTOH, if almost everyone else in the village is a straight-10 commoner NPC, with a few using the 13-STR 'guard' stats, and the local blacksmith has a 14 (because someone up thread went there, I think), 15 should do it.
Like I told Oofta, it's relative.
Your claim is that point-buy allows players to play the concept they want, and rolling doesn't
And it's true, if you take into account all the players at the table, all the time. Sure, rolling will let you play exactly the character you want, some of the time - when you happen to roll just the right stats - but it will fail you the rest of the time. Point buy will let everyone play the character they want, which may mean some of them have to curb their expectations as to what exact numbers may be involved in modeling what they want, relative to what everyone else wants.
As to 'definition of character vision', the point is that neither you nor I can claim ownership of that. The point is that each player has his own, and for your claim to be true then the point-buy method must be able to allow all players to make the concept they want.
Exactly, and that's why the strength you do grudgingly acknowledge - balance - is key. We don't all have the same vision of character concepts, the game gives us stats with which to start reconciling those visions. By putting limits and giving context to those stats, point-buy lets everyone play the character they want - as long as they don't get hung up on numbers or on concepts outside the scope of the game.
(And, it's worth noting that random generation also gives a context, the 3d6 bell-curve, and limits, or, rather, bounds, it just doesn't necessarily generate the character you wanted, even in that context, and with everyone necessarily being within those bounds.)
to disprove it I only need a single player who is unable to realise their concept in 27 points.
Again, you've resorted to a litmus test that nothing can pass. To disprove a claim about a system, you need only find one player who isn't satisfied with it? That's easy, this board is choked with dissatisfaction! Finding the player who wants to play Hercules instead of Little John in a Robin-Hood game is not hard. Jerks are plentiful.
If, OTOH, we stick to players who are going to be reasonable, yeah, everyone at the table can realize their concept - to the standards of the game and in the context of the party & the corresponding setting.
Yes, some concepts are simply out of bounds, inappropriate to the campaign in some way. If the DM wants a higher-power campaign, he'll use more than 27 points.
Much like I accept that my character can't be Superman because of game limitations, I also accept that I can't have a character concept of "Strongest man to ever walk the planet". It's not how the game works.
5e has a cap on STR, so it works fine, you're tied for 'strongest' with everyone else who has ever had a 20. It'll take you some ASIs, but you'll get there.
In 1e, you'd've needed an 18/00 you'd only have to roll up 21,600 characters (3d6, in order) to be reasonably certain of getting one.

And, you'd still be tied with a lot of other humans assuming a largish population...