According to my walkthrough guide, Pokémon with the Pickup ability will pick up Dawn Stones at level 51 and above. It gets more common the closer it is to level 100 aswell. At level 63, your Linoone has a common chance of finding Full Heals, an average chance of finding Hyper Potions, Ultra Balls, Revives, Rare Candies, Dusk Stones, Shiny Stones, and a rare chance of finding Dawn Stones and Full Restores.
IT IS RARE to find alot of Dawn Stones until level 71, but it is entirely possible. And I wouldnt complain, too much, seems a good pickup to me.
To answer your question, I dont think your Linoone is bugged or glitched in any way.