There's a really good reason the FACTS standard calls for the wire to depart the frame at an angle of at least 60 degrees from horizontal, with the intention of using two hanging points in the wall. The closer to vertical the wire departs the frame, the less stress is imposed on the frame.
The above photo shows the wire to be horizontal; departing perpendicular to the frame. That's the angle at which maximum stress is imposed on the frame. Yep, I'd bet that screw will eventually split the wood.
FWIW, corner samples for wood frames with a back-width of less than 5/8" have been removed from our displays. But when I occasionally have to deal with too-narrow moulding like that, which could happen when we re-work a customer's frame, I insist on a two-point hanging system. My preference is for WallBuddies, MagicMarkers, Beehive, or some other non-wire, multi-screw attachment to the frame, especially if it reinforces the upper corners.
If a wire is necessary for whatever reason, we follow the FACTS standard. And in this particular case, if it absolutely has to have a wire, I'd use Super Steel 4-hole hangers and double-wrap the wire ends through the loops.
No matter what sort of hanging system is used, and no matter what kind of wood frame, pre-drilling is always recommended.