I think you can't just look at the English word-by-word translation to make a point of deciding if something is right or wrong in Swedish. Different languages do work differently, and by now - having a slightly better understanding of the language than 3 months ago when I asked this question - I think it is indeed possible to say "alla väskor" (as it is possible to say "alla bilar", see Mesmes's comment).
I think it does work the same in both languages here, so that alla väskor is indeed 'all bags' and alla väskorna is 'all (of) the bags'.
I don't recognize the example with alla bilar so I can't really say anything about that, but with no context I'd say the same about cars – alla bilar should be all cars, and alla bilarna 'all the cars'.
Edit, I found the alla bilar example, it's in the Tips & Notes section. I think there's some truth in what it says there for uncountable nouns, but it doesn't hold for countable nouns. For an uncountable noun like milk, I could say either in Swedish, but I don't think I'd say all milk in English. Like Have you drunk all milk? – I'd want to add the. I'll edit the T, but it might take some time before the new version shows up.
But the examples say "alla bilar" = "all the cars", so why do we need the definite plural?
If "alla" is "everyone" why can't it be "She has everyone of the bags"?
Perhaps with a space "every one" it could already be an accepted alternative answer, if not, then next time you type it in you could use the Report a Problem button to suggest it, and the course contributors will decide. More info on every one ...
Edit: No need to report this one - see below. Thanks to Arnauti for the very fast response!
I think it would still be changing the meaning too much, because She has every one of the bags would be Hon har varenda en av väskorna in Swedish – there's a difference in meaning.