Agreed. That's actually just the head band, which is glued to the spine of the text block before the boards and cover material are set and affixed to it.

Unfortunate, but a simple repair with a glue intended for book binding (pH neutral, remains flexible when cured, etc.). I have used this product (different type of bottle [applied w/brush]) many times:

https://www.amazon.com/dp/B001I6FEHE/ref=cm_sw_r_cp_apa_glt_fabc_BHAMWBS8AZJ11SHGF1JY

Structural concerns would arise if you see some of the signatures (groupings of pages that are sewn together, which are then all sewn to each other to form the text block of a book) loosening up, falling out, etc., or the shoulder hinges giving way and separating from the boards themselves.

Yes. I used folding cardstock from Amazon:

https://www.amazon.com/gp/product/B007C6QG2G?psc=1

That specific item is no longer available but there are many other similar options.

Here is the glue I used:

https://www.amazon.com/gp/product/B001I6FEHE?psc=1

It takes some experimenting to get the right amount of glue. Too much and the card will curl. I would put a small amount on the back of the white paper and spread it out to all the corners with a q-tip. Then fix it to the cardstock. I would put a few books on top of it while it dries.