In this context, you may think that 曜日によっては＝週末には in most of the cases.
You can see them on Sundays and Saturdays, but you can't on Mondays, Tuesdays, Wednesdays, Thursdays and Fridays.
Maybe you can see them on Fridays and Saturdays. You can see customers with their children on not everyday but on some days of the week.
Answering your question, I would say:
It is because children are very busy for their homework and private tutoring schools at night on weekdays and they have no time to eat out with their parents.
So it might be, for example, on Tuesdays and Thursday when the private night school is closed.
By the way, 家族づれ or 家族連れ. Originally it meant "accompanied by," (< 連れる) which is still seen in 女づれ, it is now largely simplified to mean "a group of." This secondary use has no leader whom everyone else in the party is attached to. E.g., 3人づれ is a group of three persons, not a "leader" attended by 3 more poeple.