Well I'm not exactly sure what the oil does. I work in a B&B (breakfast cook), and do a lot of poached eggs. One of the other cooks showed me this technique several years ago. We only use double-yolked eggs, which doesn't make a difference, but the finished result is compact kind of tall bundle of poached egg, which easily slides off the slotted spoon onto the muffin.
I use the swirl method if I only have one or two I'm doing, but it gets a little hectic when you're doing six at a time!
Poached eggs are my favorite kind, I like 'em when the yolks are just barely warm, the whites are completely done, a slight vinegar taste, and enrobe the yolk. With lots of fresh pepper.