I don't think she's spoilt. I don't really think she's being unpleasant or trying to upset you, nor being particularly ungrateful either.
She's a teenager. A young one at that. Her hormones are all over the place. Her brain development is going through lots of changes. Her emotional maturity is developing still, and they tend to still be impulsive and emotional, sometimes more so than they were a little younger.
I imagine she will crave space and privacy at times. She will find these things hard to express. Young teens may look like young adults but their maturity is no where near fully grown, and they just don't behave and act in the ways we would want them too. Often young teens don't really know what they want, and they certainly can't often explain it to others very easily. Sometimes its like they are over grown toddlers. Other times like smaller grown ups.
I am sure she loves her siblings lots. It isn't about that. Its about space and privacy, especially as a teenager. 13y is right on that cusp - she will want space to be on her own, space to call her own, etc. That is normal. However, it isn't always possible for every child to have a room of their own. Sit her down and explain why. I wouldn't go down the whole "well in the olden days ..... " type route though. It never works with a child/teenager.
Sometimes I think its also okay for her to have a little whinge, so long as she keeps it to a minimum and also accepts that it isn't going to change so she needs to get on with it. Just let her sometimes have a moment to have a moan about it, agree with her that it would be nice to have a room of her own, but that it just isn't possible.
I shared with a little sister. She was 9 years younger than me. I loved her dearly. However, sometimes I would have loved a room of my own, like my brother had, like lots of friends had. I knew it couldn't be helped. I knew I was more fortunate than many other children across the world. I knew I was fortunate to have a nice home, a safe place to be, my own bed, etc. But also, sometimes I was just a teenager and craving my own space.
Is the room large enough to create a bit of a divide? Maybe using room dividers or IKEA Kallax style dividers? If so, you could split the room that way, and have hers in the more private half of the room, further from the door, where she can keep her stuff, have her pictures/posters up, have her own 'personal' space to herself. If not, could a curtain be hung round her bed area, so she can shut herself away from the rest of the world when she needs time apart?
Teenagers do need privacy and space to be on their own, and with friends. As she gets older she will also need quiet areas to work and study too. You also need to consider different sleep patterns too and if the room works for that as well, especially as both girls age - 13y and 3y will seem a much smaller age gap than 16y and 6y, or 18y and 8y.
You just need to think about how that can be achieved in some way.
I would not, however, give up my own room. No one needs to be sleeping in the living room in this situation.