If I understand your question correctly, you're asking about why OASPA and DOAJ associate themselves with MDPI, Frontiers and Hindawi. Only OASPA and DOAJ will know for sure, but I'll venture this reason: MDPI, Frontiers and Hindawi aren't necessarily questionable.
First, something to remember about Beall's list: this started as the work of one person. That means it's easily biased. OA spans from the clearly disreputable on the one end to a very gray area on the other. Beall undoubtedly had good intentions, but if the Who's Afraid of Peer Review? sting meant anything, Beall was only 82% accurate. In the sciences, a theory that predicts the right result 82% of the time is good but not great; in particle physics we even need a 5 sigma result (p-value 1 in 3.5 million) to claim a detection. I'm not saying Beall was wrong about MDPI, Frontiers and Hindawi, but I will say that "because Beall said so" is not a sufficiently good reason to conclude ____ is predatory.
Now about each publisher:
MDPI: See Wikipedia for more information. You can see Beall's criticism of MDPI stems from several aspects, such as how MDPI's articles are lightly-reviewed, how MDPI uses email spam, and how MDPI listed Nobel laureate Mario Capecchi on an editorial board without his knowledge. However:
- Many OA journals do indeed review lightly. For example I once attended a talk by a Springer spokesperson who talked about a journal which reviews for correctness, not novelty (can't find the journal now, but PLOS ONE has the same policy). Viewed one way this is laudatory - it makes peer review less random by eliminating one completely subjective facet! Viewed another way, this is terrible - it makes it seem as though the journal will publish old results known for hundreds of years as long as the author is willing to pay. Which is closer to the truth? You'll have to come to your own conclusions.
- Email spam. Although everyone finds them annoying, what constitutes email spam isn't universally agreed on. If you receive an email from someone you don't know with "Dear Professor Strongbad, I saw your question on Academia.SE and find it interesting, would you like to write an editorial on predatory publishers for my journal" - would you call that spam? Some people would, others would not. Also, what exactly isn't email spam anyway? If you never emailed people you didn't know personally, you would never be able to expand a journal large enough to be self-sufficient.
- Finally the Mario Capecchi case was later shown to be the result of inaccurate communication by Capecchi's assistant.
Frontiers: again, see the Wikipedia article. You'll note that, similar to MDPI, there were established academics who defended Frontiers. Although the volume of allegations against Frontiers in the article is both larger and harder to justify if true, it's also the case that a Frontiers journal rejected John Bohannon's sting paper. OASPA and COPE both investigated Frontiers, and both decided that Frontiers meets their membership criteria.
Hindawi: once again see the Wikipedia article. I don't want to rehash everything I wrote about MDPI and Frontiers since a lot of it also applies to Hindawi, but I'll add a few specific things:
- Hindawi was one of the pioneers of OA. In 2007, they converted all their journals to OA - this was both 1) before OA really took off and 2) pioneering, since even today most big publishers don't use a complete OA model.
- Hindawi is big. With over 400 journals and tens of thousands of published articles a year, Hindawi is a big fish in the OA pond.
- A Hindawi journal also rejected John Bohannon's sting paper.
- Some of Beall's criticism of Hindawi apparently focused on how high its profit margins are (apparently higher than Elsevier's). This not only has no relation to the quality of Hindawi's editorial process, it's also the case that Hindawi's article processing charges are lower than average, and they're based in Egypt, which as a developing country has much lower labour costs than the Netherlands-based Elsevier. One could say that Egyptians are bad at publishing relative to the Dutch, but that's borderline racism.
tl; dr: it's not a given that any unbiased observer will conclude that these three publishers are disreputable. Accordingly, it shouldn't be surprising that some OA consortia are willing to count them as one of their members.