14 of the “no dominant order” languages are double-marking
11 of the “no dominant order” languages are head-marking
7 of the “no dominant order” languages are dependent-marking
1 of the “no dominant order” languages has other marking.
4 of the “no dominant order” languages have no marking
That makes it appear marking the head is more important for freeing up word-order than marking the dependent is.
But although 25 of the 37 NDO languages in this list do mark the head, still 21 of them do mark the dependent.
So to free up word-order it’s important the language mark something
. That’s a bigger deal than it is to mark the head.
Of course case-marking is a kind of dependent-marking.
11 NDO languages without cases
5 NDO languages with “exclusively borderline case-marking”, whatever the heck that means
1 NDO language with 2 cases
3 NDO languages with 3 cases
2 NDO languages with 4 cases
2 NDO languages with 5 cases
4 NDO languages with 6 or 7 cases
3 NDO languages (including Wambaya) with 8 or 9 cases
6 NDO languages with 10 or more cases
So of the 36 NDO languages here, 16 have no cases or “exclusively borderline case”, while the other 20 have cases.
Among the 20 that have cases, the median number of cases is about 6 or 7, I guess.
This is not sufficient data from which to draw a conclusion. I, or we, would also have to see whether knowing the number of cases helps us predict the word-order; or, at least, whether the presence of cases (or of more than three cases) helps us predict whether there is or isn’t a dominant order.
My battery is getting low, and it’s getting late, so I won’t do that now.