I have published a few papers with MDPI also (Sensors and Sustainability journals). I have also served as a reviewer (and received a discount for OA fee). In my opinion the peer-review was always standard. I even had a rejection in Sensors. My foreign colleagues (including such “aces” of science as Steven S. Ripp) consider Sensors journal highly and are not afraid of publishing with MDPI.
I am not able to evaluate this paper, it is beyond the scope of my expertize. Nevertheless, also with experience as a associate editor for “standard” Springer journal (International Journal of Environmental Science and Technology), I can set at least two hypotheses why author might be interested in publishing with MDPI (and not mentioning “they publish everything”):
1. The review process is fast. This is general not because of poor and too-fast reviews (MDPI requires the review to be finished in the same two weeks as our journal) but because of a professional editorial board, who takes care of submissions as a full-time job. On the other hand I work for university, teaching, resolving projects, supervising theses, managing department and as a volunteer bonus, usually in the evening or weekends, checking new submissions, inviting reviewers (of which cca 50% do not response at all and 30% decline to review), reading reviews etc. Typical submission is “sleeping” 1-2 weeks until editor-in-chief assigns it to associate editor (me), then another few days (sometimes even week) until I carry out preliminary check, review anti-plagiarism report etc., then another 1-2 weeks until at least two reviewers agree, the 2-3 weeks until reviews are finished, a few days before I check reviews and submit decision, and few days (sometimes up to two weeks) until editor-in-chief confirms it. In sum 2-5 months (often more) elapse until the first decision is reached. The professional editorial boards in MDPI reduces the idle (editorial) times to minimum and the decision is usually reached within month or two.
2. If the author, unlike Mr Beall :-) is a fan of open-access (general readability is a significant advantage if you want to spread your results especially outside scientific community) than MDPI is definitively better choice than “gold” journals offering open-access option. Elsevier, Springer, T & F and other big publishing houses (and also highly profitable, among others because of “paraziting” on volunteer work of us editors and reviewers) offer the OA option for 2000 EUR / 3000 USD (and for those OA papers the publisher receives the money twice, i.e. from authors as well as from subscribers, who subscribe entire journals or journal sets, not single papers). MDPI OA fees are much cheaper, even for journals indexed in WOS or SCOPUS.
By the way I suggest to check the quality of Springer-Open journal. Their editorial policy is such that novelty is no considered if the text is sound.