As far as I know they were put whole, probably already dead though.
Whether for using as fuel or for rendering their fat, they were not butchered to get only the useful parts and were processed whole instead (which is why the oil was often contaminated with feathers as the article says).
But I wasn't there then of course, and it's not described in "15,000 Miles in a Ketch", one of the best accounts of the era that I know of. Maybe it's better talked about in "Narrative of the Wreck of the Favorite on the Island of Desolation" because the author, John Nunn, was taking part in a more "industrial" seal harvesting operation but I haven't read it yet. In any case I recommend reading at least the first of these books, and the second book is likely interesting as well if one is curious about that era.