You have noticed a very peculiar aspect of English vocabulary. As rich as it is in comparison to many other languages, due to its almost creole history, it really is impoverished in comparison to other languages in kinship terms.
But 'why' is always a difficult question, especially when mixed with cultural questions. There are the difficulties with Sapir-Whorf type explanations: both language restricting thought on one hand and the number of Eskimo words for snow on the other.
Does the lack of kinship terms reflect the cultural lack of warmth and caring for relatives among English speakers, that is not caring leads to the loss of the words (which etymologically do exist in the ancestor languages), or did the arbitrary lack of kinship terms contribute to the crumbling of English family values?
Any direction sounds much too tendentious, too judgmental, and requires too much unjustified and biased assumptions to choose.
The comparative lack of kinship terms does ask for an explanation but one backed up by linguistic and anthropological and comparative research. The only source that comes to mind is Levi-Strauss's 'The Elementary Structures of Kinship.' (primarily anthropological but as a by product a number of examples of kinship term systems.
English isn't alone in having relatively few kinship terms. Some other European languages have only a few extra (French, German) and some languages really only have names for their clan and generation (anybody of one's biological parents' generation might be called something like 'uncle' or 'aunt', even one's birth parents).
Having no definite answer to your question, I can only say beware of making cultural inferences based on restrictions to languages. Some languages have grammatical gender and others don't, but that doesn't mean the ones without can't recognize the sex of other people.
Best Answer
Notice how when listing out all three siblings, it’s perfectly fine to use triplets, yet when it comes to speaking of just one or two of them alone, this feels “somewhat intuitively wrong” to you.
The difficulty you’re having with triplets compared to twins is that it’s hard to mention someone’s twin in a way that leaves the other twin completely out of the picture, but with triplets this is quite easily done and this may feel odd. Your intuition fills in the missing member of a pair of twins, even if it is someone unnamed. It’s harder to do that with a set of triplets.
Nonetheless, the use you're not comfortable with can be found in published writing. It is rare, and most of it is recent works of fiction. Here are a few examples.
Cerea – Adventures in an Online World; Weby; 2007. [reference]
A Sorrow in Our Heart: The Life of Tecumseh; Allan W. Eckert; 1993. [reference]
Children of the Unndis Sedna; Lynelle Souleiel; Archway Publishing, 2014. [reference]
Dead Forever; Willian Campbell; 2014. [reference]
Terry Spear's Wolf Bundle: The Heart of the Wolf, Destiny of the Wolf, and To Tempt the Wolf; Terry Spear; 2010. [reference]
Touch Me in the Dark; Patricia Rosemoor; 2011. [reference]
Priests Under Pressure in Southern Moravia: History and Identity in Roman Catholic Polemics (1675–1615); Adam W. Darlange; dissertation thesis, 2009. [reference]
It’s possible that the comparative rarity of triplet births is contributing to the unfamiliarity of the term. Twins number one in 30 natural births, but triplets are only one in 1,000 – roughly the square of the frequency for twins.
If you’re not comfortable with it, than don’t use it. Just know that some people do use it that way.