On mobile you have to either
a)be level 85+-95+ depending on difficulty
b) buy the crown which requires the primal pass and 100+ amber
c) have your name end with a surten syllable (look on yt/Google for the options) and be level 40+
If you don't do that then the Griffin won't eat when unconscious.
Sorry for th long tip.
If you have access to a snow owl, you can tame these with any crossbow! Just shoot it until it looks bloody, and heal it with the snow owl’s ability. Do it so that it heals quite a bit, but not so long that the torpor drains back empty. Then hop off the owl and continue shooting it with tranq arrows. I knocked a 140 with a 279 crossbow and using this method it had over half hp remaining once it finally went down. Help this get to the top!