Why not the family are going? Group nouns can usually be either and there's no context here?
Present Simple "goes" translates to "chodzi".
"idzie" happens right now -> "is going".
Couldn't it mean "the family goes to the restaurant" i.e. they do not eat at home?
If the family "goes", i.e. does it with some regularity, it should be "chodzi".
Why does "my family is going to the restaurant" not work?
I got the impression from a comment on a previous question, that when talking about family it was assumed to be your own?
Well, the subject's own. And here "family" is the subject.
Sure, sometimes also when it's the subject, but then usually it can work like that in English as well. For example "Tata idzie do restauracji" will be understood as "My dad", but in English you can also say "Dad is going to the restaurant" and it also will be understood that way.