Is "to take birth" correct English?

No, a Western English speaker would say

She gave birth to a healthy baby boy.

He was born with a full head of hair.

The sense of "take birth" in English in the 19th century was that of arise, or take hold. From a book of songs published in 1822, the following oh-so-British celebration of pleasure:

From prudence let my joys take birth,
Let me not be passion's slave,
Approv'd by reason, sweet's the mirth,
Vice of pleasure is the grave.

Another example from Andromache; or the Fall of Troy, a tragedy by Thomas Paine (no, not that Thomas Paine):

Thy dreadful thunder o'er the brain when first
Imageries of guilt, confusedly rise,
As toads engender, 'bove the darkling stream,
To take birth in the sun.

This reflecting the old notion that toads arose by spontaneous generation from the action of sunlight on mud.

When the British confronted the religious practices of India, including sati, Thuggees, and the car of Juggernaut, they turned their minds to bring the "enlightenment" of Christianity to the "mythology" of the subcontinent. Missionaries wrote proselytizing pamphlets with passages like this one:

He [Jesus] said, " I "will take birth on earth, and suffer all the " torment of sins on my body. Whatever " sinners take refuge under my protection, " you will grant them salvation." God said, " Yes, this is my promise, I will redeem " them.

I think the language here is an attempt to phrase Christian apologetics in the local religious language. It may be found quoted here in 1803, though not with approval.

In American and British English today, the will governing birth is seen to be that of the parents, not the child, so mothers "give birth," and children have to take what they get.