I tamed one today, and it was really easy.
Dont care about to build a cage or something. They goes down after a few arrows and are really slow, so just walk away from it and shot a few times. It was like four prime meat whitout any narcotics. Dont worry about to tame new dinos, its just 99% of the times it goes bad as hell.
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.