It can be reusable and yet unique on each page. And those should get id's, and are not rare at all.
> The percentage of things on a page that are "singletons" is very small compared to reusable widgets.
That isn't necessarily true. Obviously there are different types of pages, but a typical page will have a number of elements on it that are each unique. (They might appear on more than one page, but on any particular page, they are unique.)
Give them id's, that's what they are for.