I am guessing that the small size is actually just the result of abstractiong/ simplifing things due to technical and timewise constrains.

Wasn't the strong radiation everywhere after the bombs fell a huge problem for the people and only very few communities actually survived without become ghouls. I can only remember one example where someone survived outside of the vaults and that was in honest hearths.

No it literally says on in blueprints on the walls what the maximum occupancy was. For example Vault 81 was 96 people.

Well yeah but not everyone died because of the bombs, for their to be any population at-all we must assume that plenty of people survived to reform in the post-apocalyptic wasteland. We know of plenty of people, the Hilltop family in Fallout 3, the Miller family and associates in Fallout 4.

And Randell Clark talks about a lot of people who survived the nukes and made it into the canyon. There is also Raul even though he is a ghoul talks about people who survived the bombs. Las Vegas/Mojave area was for the most part saved by Mr.House.

But yeah people survived the nuclear war in other locations then Vaults.

Vault 114 has a max of 120. Still I agree with you that is a very small number of people for a Vault. The other Vaults have less then that.

True but that was more of an expetion. I remember reading that the following nuclear winter and the spread of the fallout were a bigger problem for the survivors than the initial blast. Not to mention the fall of civization.