I think you're missing a key part of citing (at least for me), paraphrasing. When discussing sentiment or similar without quoting, you wouldn't know what to search for. It makes sense to give a page number to review and get more information on the source, without that you wouldn't know where to look. You'd have to read the entire source.
For text, only section in your "less-granular searching" would work, and not necessarily particularly well. Page number would often be more granular, and that's what's needed for paraphrasing - exactly where to find what you're discussing - but not exactly what it said.
If you're paraphrasing, then you could direct-quote a critical part of the text so that a reader knows what to search for.
A writer could also adopt page-neutral paragraph or sub-paragraph numbering. This is common, for instance, in classical texts (where such numbering has been done retrospectively, by scholars, because of the difficulties of citing passages with so many translations - the same general problem that paraphrasing has). To cite a few major examples, Plato's and Artistotle's works have such numbering, as does Hobbes' Leviathan.
Page numbering has poor granularity for citations in real scholarly work (where you'd ideally want to be able to cite within a few lines at most), and it prohibits flowed text. If someone has some case where they think they need page numbers for citations, they should figure out an alternative (a few of which have been outlined above).
For text, only section in your "less-granular searching" would work, and not necessarily particularly well. Page number would often be more granular, and that's what's needed for paraphrasing - exactly where to find what you're discussing - but not exactly what it said.