「たとえ + ても」 vs just 「ても」, what is the role of たとえ？ Is it just emphasis or something more?
You could possibly label the usage as "just for emphasis", but without 「たとえ」, the contrastive conditional simply would not sound "complete" or even "right". If the other native speakers here disagreed, I would like to hear from them.
That is because, according to me, being emphatic is an essential part of the contrastive conditional to begin with. Imagine eliminating the word "even" from the English contrastive conditional sentences using "even if". Would the sentences still sound the same way without "even"? I would doubt that very much.
Thus, I would suggest that you learn 「たとえ + ても/とも/としても, etc.」 as a set phrase.
Nothing to do with the question, but you will occasionally encounter the word's alternate spelling 「たとい」.