First of all, I heavily disagree with balance only mattering in multiplayer, this game is supposed to be strategy even in single-player and any desire to go farming/rpg/simulation can be already achieved with lower difficulty. Second of all, it's not just matter of balance, but gameplay consistency. Even If you disregard both of these as matter of taste, comparing it to UD is very wrong, as not every civ has UD, but every civ has something it its place that should be equal in power or the civ is compensated. Even "civs" that have extra unique have it as part of their power, even stressed more by Firaxis including it in their tooltip every now and then when the tech/civic prereq is different (such as Tagma being part of Basil II's power budget, not just some random extra).
If that's the way you want it, then yes, you could weaken the civ mildly to include unique resource as part of their power, but I think it ends up being too underwhelming to be desirable. That way it's better to have one civ that has such unique ability that they have several unique resources they may trade away - If you want the civ to be the master of trade deals. Maybe it's matter of taste, but If I were choosing civ I would find their unit, district, building or improvement far more interesting than resource. On the other hand If you don't want it to be generic luxury but for the resource itself to have some bonus, the difference being that the civs can willingly lose the bonus and give it to someone else in trade deal, while I still consider it underwhelming, I would at least with this agree it could add something interesting to the game and I could see why it would be appealing to others, that I admit.