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.
Lure an argie into an enclosure where it cant escape and you can ko it or tranq the argie till the argie starts running away then pull up with a moth and fart on it and then its basically stuck for 30 seconds, easy ko
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.