I've never done a scarf in the round but the other knitter in my family has done several. (She can make an intarsia colour pattern for a sports-team logo and not worry about the messy back.)

One purl stitch on each side helps them to lie flat, although interestingly they don't choose to fold exactly on the purl.

She just sews up the ends, but you could also graft if you like grafting.