You are mostly quoting what Chinmaya Mission has popularized. When I argued with them, all they said was it was his mother’s maternal home and his birth there is an assumption based on tradition. If oral tradition says Kalady, I would go with that.

Veliyanad was the ancestral home (illam) of his mother, Aryamba. Under the traditional marumakkathayam matrilineal customs followed among Nambudiri Brahmin families during that period, pregnant women commonly returned to their maternal ancestral homes for childbirth and the early months of child-rearing. Accordingly, Adi Shankara was born at Melpazhur Mana.

Kalady, on the other hand, was the ancestral village of his father, Sivaguru. After the initial months following birth, Aryamba and the child traditionally moved to the father’s household in Kalady, where Adi Shankara spent his childhood and undertook his early gurukul studies. This explains Kalady’s later prominence as his hometown and spiritual center.

The original birthplace, Melpazhur Mana in Veliyanad, still exists today and is preserved as “Adi Sankara Nilayam” under the management of the Chinmaya International Foundation. It remains a highly revered site associated with Adi Shankara’s birth.

Suggested correction:

Birthplace: Melpazhur Mana, Veliyanad, Kerala
Childhood home and early education: Kalady, Kerala