Bibliophiles have their own jargon for the various ways a book can be damaged— foxed, warped, a whole slew of c- words from to chipped to cockled to crimped; see ILAB glossary for a few. The technical term for a book that has lost its binding is disbound, which Merriam-Webster defines as simply
no longer having a binding (a disbound pamphlet)
But there isn't, so far as I know, an English idiom that refers to a book becoming disbound due to overuse. For whatever reason, the English noticed wear and tear on the pages of the book rather than its spine, and so a book that is worn out from overuse is very often described as tattered and/or dog-eared, and perhaps frayed (especially of cloth bookcovers). Per MW again:
tattered - 2: torn into shreds: ragged (a tattered flag)
3a: broken down : dilapidated (decaying houses along tattered paved streets — P. B. Martin)
b: being in a shattered condition (led their tattered party to victory)
dog-eared - 1: having dog-ears (a dog–eared book)
2: shabby, timeworn (a dog–eared resort dog–eared myths)
(a dog-ear is a folded-down corner of a page, used to hold a place)

3 fray (intransitive) - 1: to wear out or into shreds
2: to show signs of strain