I used a tranquilizer bow and my partner use long neck we took turns shooting as it was hingry for me. Once it was KO we supplied it with raw meat and no berries or juice made a saddle right away and in my opinion best flyer so far
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.