А почему это не учат в учебниках английского языка? Его даже президент США, Хилари Клинтон использовали в своей речи

The article "a" has two pronunciations, a weak form "a" and a strong form (ei).

The strong form of the article "a" pronouced (ei) is not used often.

When used, it is used emphatically--
1) to show contrast
It is a (ei) solution, but not the only solution.

2) when stressing the indefiniteness of something.
Question....I lost my wallet. Have you seen it?
Answer....I saw a (ei) wallet, but I don't know if it was yours. The strong form emphasizes the indefiniteness of the article "a".

3) to stress singularity--
I have a (ei) cat and a (ei) dog. I have one cat and one dog.

4) to name a word--
This animal is referred to as a (ei) platypus.

There are no strict rules for the use of the strong article a (ei). It's use is a conscious decision. Some people consider it's use incorrect or an affectation.