The Comments section of that page has the explanation:
Wilson & Reeder (eds., 1993) note that "the pinnipeds (otariids,
odobenids, and phocids) are included within the suborder Caniformia;
placing them in a separate Order would make the Carnivora
paraphyletic." This arrangement continues to be followed in Wilson &
Reeder (eds., 2005)
If both Carnivora (Bowdich, 1821) and Pinnipedia (Illiger, 1811) are orders in the traditional Linnean taxonomic sense, then one cannot be nested within the other. If this were to happen, then Carnivora would be paraphyletic because it would exclude some of its members. The same argument is made when discussion birds as dinosaurs (you can't talk about Dinosauria that doesn't include Aves).
The problem arises when phylogenetic systematics (clade names) meets traditional Linnean systematics (orders, suborders, etc.). Phylogenetic systematics does not distinguish these higher taxonomic groups. Clades are just nested groups of monophyletic taxa.
So then to answer your last question, Pinnipedia does appear to be in common usage (Google Scholar reports several hundred citations in the past few years). The key is to distinguish Pinnipedia as a traditional Linnean order (not valid) and Pinnipedia as a monophyletic clade that includes seals, walrus, and sea lions (perfectly fine). Seals, walrus, and sea lions are all more closely related to each other than to other Carnivora, so they are monophyletic, and the name for that grouping is Pinnipedia.
Here's the Tree of Life page that has all the clade names.