They​ recipe book will attempt to use damaged bows when crafting dispensers. Currently only undamaged bows can actually be crafted into dispensers, although named and enchanted bows can be used.

I don't know if it is intended to be able to craft dispensers with damaged bows (personally I vote yes please), but obviously either the recipe book or the actual recipe is wrong in this case.

Note: The unenchanted bow on my hotbar is named; hence why it is ignored by the recipe-book - expected behaviour

sidenote: I like the mechanic of named/enchanted bows being usable to craft dispensers but not automatically placed into the crafting grid by the recipe book - I feel this may prevent unwanted consumption/destruction of players' “good” bows