He hesitates at that, studying Wrathion. It seems like the options are go with him or don't go, there's none where Wrathion stays and rests his leg.

"... Are you sure you should be walking?"

There's also the other aspect -- he'll need other tools too, and Wrathion can already smell the blood on him. He's likely to be more upset once he's helping him dig bits of arrow out of himself. That is unavoidable, however unless Anduin somehow finds a way to send Wrathion out while he does it with a mirror.