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.
I'm glad that you're at that economic stage that you're worried about guests and decor
if they care about how your private space is set up, they don't need to be there
Take me, for example. I'm judgy as hell but I'll never get an invite to your bedroom and that's totally okay. Assholes like me don't need to be over to your place making all these judgements. Your friends and lovers will be glad for your company and it's all good.
Also, when I was a kid were were too poor for carpets to cover the rough flooring under some rotting carpet we pulled up, so to reduce the splinters from walking around someone gave us a bunch of blue latex paint that kinda sealed the wood a bit as it coated. We were definitely 'no headboard' poor but we got better.
I like mine cause it stood my pillows from falling if the bed gets pushed or slightly, it's also padded so when u was depressed and spending 18 hours a day on bed it was nice to sit against.
I wouldn't be opposed to not having one though. As others have said, it kind of depends on the style you're going for.
Honestly they seem out of fashion nowadays. It's one more thing to move when you move, it cost a lot for very little, most people lack a tool set to attach it to the bedframe.
Headboards typically come with the right size allen wrench for its bolts... then you just need something to hold the nuts; wrench, pliers, non-stick pad etc.
If you're worried about appearing cheap, to people who visit your house, particularly the type of people that might judge a person for appearing cheap, and the opinion of that type of person matters to you, then yes, buy a headboard. Or, if this is like a survey and you're asking for everyone's individual opinion, I personally would not judge you as "cheap" for this, but everyone's threshold for these things is different.
I bought a padded, fabric covered Chesterfield style one on eBay for about $100 AUD. It's comfortable to sit up against and drink my coffee in the morning. I like the aesthetic but it's definitely also a comfort thing. Cold hard wall or pillows falling down the back doesn't sound appealing.
When I've slept on a bed without a headboard, my pillow tends to fall off the end of the mattress as I push it around when I sleep. The headboard acts as a fence to keep the pillow on the bed. To me, a mattress sitting on a box spring held up by a frame doesn't look so much cheap as incomplete. But since the bed is typically out of sight in a bedroom, people generally don't sit around gazing at it.