I suspect that if we venture into the streets of an American city and interview random passers-by, fewer than 10% will know that "chai" is a word that already means "tea".
In this case, I would argue that it does. "Sahara" means "desert", but "Sahara Desert" is still the proper name for that place. "Naan" means "bread", but it's also the name we use for the kind of bread originally made by the people who speak the language where "naan" means "bread". Likewise, "chai tea" is a perfectly fine name for tea made in the style of the people whose word for "tea" is "chai". Languages do this all the time.
It's great to refer to things by their proper names, never mind the corruption that English has adopted.
Still, rude to correct someone for using what's essentially a colloquialism. If enough people use a word 'wrong' then, well, that's a new usage that we can't ignore or get prescriptive about. Because prescriptive usage is from some point of view after all. The Chai tea people have their point of view, and are correct within their community.
Except you're wrong, and this type of construction is extremely common. Take "pine tree" for example. There's nothing "incorrect" about calling it chai tea.
You're not a purist, you're just incorrect. We are speaking English, so saying that it is redundant isn't true. Just like how Hound means generic dog in German but hunting dog in English, or rouge means red in French but means a kind of makeup in English, Chai doesn't mean tea in English and thus there is no redundancy here.
Chai means (Indian) tea with milk. It’s a borrowed term in English that already includes the meaning. Just like you don’t use “yoga exercise” in English because yoga already means a form of exercise. Or the usage of “karma” in English. You don’t say “karma fate”.
Yes, loanwords don't operate consistently when brought from one language to another. That does not mean that one way they work is "wrong" and the other way is "right", it just means language is complicated. You're the one coming here and telling people they are wrong.
Edit: Like in Italian Latte just means milk, but in English it means an espresso with steamed and then frothed milk on top, what would be called a cappuccino in Italy usually. Americans calling it a latte aren't wrong, they're just using a loanword to mean something different then the original language.
How is chai tea used differently in America then? If I ask for it in a coffee shop, I usually get chai (tea) with steamed milk, which is exactly what the original usage is.
Looking at the Starbucks app right now for example on their menu: Chai tea is black tea with clove, cardamom, cinamon, and ginger and no milk by default, Chai Tea Latte is that black tea and spices with steamed milk. You'll also see Chai sometimes to refer to the spice mix sold without any tea.
Chai (or a variation) is the name for tea in half of the world [1], so if you say "chai" you are just saying "tea", while if you say "Chai Tea", if I understand correctly, you are probably referring to a specific beverage [2]