You can ditch the off-screen rendered pages but you are still stuck with the entire 1000-page data structure. Should your current line be on page 997 or 998? You have to calculate the size of every line of every page to answer that.
There are numerous data structures that are well suited for various geometric queries like ranges/lookups (interval trees, quadtrees) as well as more text-oriented operations like cut/copy/paste/insert/merge/etc (like ropes).
I'm not familiar with the operations required to put a cursor at the right place in a document, but knowing how much research has gone into storing similar data and looking up what you need efficiently the idea of "going through all the text every time" is a big code smell.