One thing to keep in mind is that we're used to bigger housing these days. Smaller housing in the wasteland makes sense because it takes less fuel to heat and lets your family huddle in the dark in comforting togetherness while the supermutants are out howling at the moon.
As for a vault, every inch is precious (not to mention, if you've got an efficient design, 1x1 is actually more than enough for a bedroom when building, bed against the back wall, side table on one side of the door, shelf or small dresser on the other side. Candle on the side table, some pictures on the walls)
For examples of efficent space usage, look to the RotC County Crossing plan, second floor. There's a full bathroom and a kitchen up there that are each in a 1x1 square and both appear reasonably functional.
In many ways this is true. although you're missing a few things.
Realistically, human beings weren't really meant to live underground. Humans like to roam in open spaces. So for humans to be truly happy in vaults and not go crazy from cabin fever, it would need to feel spacious, etc and have every inch of the space utilized well.
on the outside, people can have small homes because they're often outside of it going to work, socializing at the bar, etc. they get to be outside in the wide open space and then come home to huddle up with their families by the fireplace. To the people on the surface, a tiny house is just where their bed and a warm fireplace is. not much else is required, actually.
Inside a vault is a whole different story however. They're stuck there inside all day, all night. Cabin Fever is no joke, it can pretty much drive a person insane. In fact in real life if vaults existed this would be one of the reasons why vaults wouldn't last very long.
So yeah, you have to make some areas of the vault roomy, spacious and also give them plenty to do... and also there needs to be new stuff to do on a regular basis so that they don't get stuck in a rut, and run risk of suffering from Cabin fever.
That is, if you're an overseer who genuinely cares about their residents and want the vault to last a very long time. Otherwise they probably wouldn't care that their vault feels claustrophobic and stifling to the residents after a long while of living there.