It isn't hard to tame them in the wild but it also isn't easy. Lure the bird you want to tame to a taming area or pen. starving it like anything works best. Got mine or four prime meat. Level up stamina and health. Can't level up movement speed on PS4.
An argentavis can't fit through a ceiling with one tile missing, but a pteranadon or lymantria can. Build a trap with one corner ceiling missing, aggro the argentavis, fly in one side and out the missing ceiling tile, and land. Dismount, run around, and close the entry door. Tame safely at your leisure.