First he argues that metatherians were more severely impacted by the mass extinction event at the Cretaceous-Paleogene boundary. Crown marsupials (the living species and their common ancestor) post date that event. In contrast crown placentals originated in the Cretaceous. (For Metatheria vs. Marsupialia and Eutheria vs. Placentalia please refer to my previous post).
An ungulate Theosodon garretorum and a carnivorous metatherian
Borhyaena tuberata from the Santacrucian (Early Miocene) of South America
Second the fossil record does not support the notion that one group of mammals out-competed the other. In the Late Cretaceous, for example, metatherians were more abundant than eutherians in the North American fauna. In South America, eutherians such as the native ungulates went extinct, whereas metatherians including crown marsupials survived. (Most of the present fauna, including rodents and Neotropical primates, arrived later, as discussed here.)
Third the metatheria that did survive into the Paleogene were (with very few exceptions) confined to the Southern continents of Antarctica, Australasia and South America. Major clades of placentals that show a comparable distribution are Xenarthra (armadillos, anteaters and sloths) and Afrotheria (including elephants, hyraxes and tenrecs). There are 35 extant species of xenarthrans and 83 afrotherians against 340 marsupials.
Thus the discrepancy in species richness between marsupials and placentals is due to the success of the remaining clades, Euarchontoglires and Laurasiatheria, that evolved mainly in the Northern continents and reached Africa and South America much later.
Perhaps the chorioallantoic placenta did give the edge to placental mammals. But Sánchez-Villagra argues convincingly that even if developmental biases exist those constraints can be circumvented. They cannot fully explain why there are 15 times more placentals than marsupials in the present day fauna.