Not sure what their criteria are because
Preston,
Bruno,
Jameson, and
Ryder are definitely not on the brink of extinction... In fact, they're at peak popularity right now.
Leroy and
Buddy I would agree with. I'm not sure
Tiger was ever really used to begin with.
Boy names actually on the brink of extinction:
ChesterDeweyDonaldElroyGeraldHerbertHowardIrvingLesterLorenRayWilburWoodrow