Essentially a travel document with rights for some consular assistance, its usefulness seemed limited to many for anything but easier access to the oversea travel.

划线部分为 seemed limited 的目的状语。

many = many people，to many 对许多人而言。

由于谓语部分意思是“受到限制”，带有否定意义，所以，anything but 的意思，应该反过来按 nothing but理解：除了方便他们海外旅行外，其他的不可以。译为：

但对许多人来说，它的用处似乎仅限于方便他们海外旅行。

注意习语表达：

anything but A：除了A 外，什么都是；决不是A。

not...anything but A  = nothing but A ：除了A 外，什么都不是；仅仅是A。

He could speak anything but Greek.

He couldn't speak anything but Greek.

He could speak nothing but Greek.

他不会说希腊语。

他只会说希腊语。

他只会说希腊语。