I've seen teams do that in OT because possession is so critical and it's relatively easy to maintain in open ice. For instance, the Bruins would start Mark Kastelic to win the face off and as soon as they gained control the defenseman, usually McAvoy, would do a little figure skating around his own end while Kastelic would charge to the bench.
There's also no law that says the center has to take the draw. They could put Lodewyk on the wing but have him take face-offs, particularly those on his strong side. Lang said in one of the podcasts that he particularly likes to recruit centers and move them to the wing if necessary as they've had to develop a more robust (my word, not his) offensive and defensive skill set.