Part of it is that you know the other parents want their child to be home for their own dinner. You don't want to wreck their plans. As I said each family tends to view dinner as an important get together time.
Another aspect is that we Scandianvians are quite private and slow to get to know. We don't like to impose ourselves on others. You got to understand that this setup works both ways.
I think when many foreigners read about this, they think there is this child who eagerly wants to join the family dinner but who is denied it. But reality is that very often the other child is very uneasy about joining the dinner of another family.
It is just in our mentality. We don't like inserting ourselves into the private sphere of other people. It is like when somebody blocks you in the groccery store, we Scandinavians usually don't say "excuse me." Why? Because anything that feels like imposing on others is something we dread. We rather walk half-way around the store to get to where we need to be than to ask somebody to move. Or we may simply stand and wait until the person decided to move.
Thinking about myself when I was a kid I think I would have been really uncomfortable sitting and eating dinner with another family. It is their private time and I am putting myself in the middle. Because you are new person at the table you put attention to yourself.
That is another Scandianvian phobia: Putting attention on yourself. We tend to hate that.
We have kids over at our place frequently. And we actually ask at times if they want to eat with us, but none of them wants to. Kids are a bit more shy and reserved that probably an American or Italian kid would be. We are however not a very typical Norwegian family. My wife is Asian-Canadian and we don't eat quite a regular Norwegian dinner time hours.
Kids are often around and about in the kitchen when we eat, so we may be eating while talking to them. So it is not like they are excluded. I realize to an Italian this sounds really strange to have people around while you are eating, who are not doing it but generally they don't want to. The social convention is that you eat at home with your family.
It is of course different if you have specifically invited them for dinner.
Different families view these things in different ways. For some Scandianvians food is a pretty function thing. It is like dong a chore, so why would you demand that other kids carry out the chore your kids have to do?
The other is a privacy oriented culture. Scandianvians tend to love alone time and small groups. Lots of people love going out in the forest alone to be with their own thoughts. Crowds of people stress out Scandianvians. Gathering crowds of people for social activity, especially if it is spontanous kind of goes against Scandianvian mentality.
I am quite fond of people myself but I can only deal with lots of people for so long before I need a time-out and spend a day for myself. Time with just the family like a dinner is kind of a way of stressing down. Just having the closest people you know well is calming.
I don't know if this is really hard to relate to. People are different. I have met however people from socially active countries like Brazil or the US who are more private and shy and who love living in Scandianvia. The pressure to be social in Brazil or the US stressed them out. My wife grew up in a very extrovert culture, the US, and always found it stressful. I am actually much more of a social person myself so I would not mind living in a more socially engaging country than the Scandinavian ones.
But my wife loves it. A Brazilan friend of mine remarked on how he loved that nobody showed up unannounced. Personally I thought what he described about Brazil was a bit nice.
Sorry this is a bit rambling, but it is like trying to think out aloud about how us Scandianvians feel about this thing. Why do all this stuff that looks and feels so alient to foreigners seem so natural to us. I hope it helped make some sense even if I am not able to be entirely coherent.