My bed didnt have a headboard. Nothing wrong with that.
I will say its kinda nice to have but plenty of beds dont have it. I'm sure some people see it as odd or cheap but who cares what they think? Its my bed. And a headboard can be added to most frames. Hell i used 2 ikea tables mashed together as headboard for a while.
You can buy ones that just stick to the wall. Honestly, I don't want my head rubbing against the wall and leaving grease marks while I sleep, or fingerprints. You might not notice them but you definitely will once you move the bed.