I wonder whether anybody has developed a CLI browser that also has a JavaScript engine. I've seen plenty of TUIs that have modals, text line inputs, etc. You could probably get pretty far so that at least you can support more of the Web. At least to start with just using an engine to allow JavaScript to have control over HTML elements would be a start that would solve many of the issues with text browsers.
Some CLI browsers have JS support but what's missing is up-to-date support. The existing browsers have way too much technical debt so adding something like TUI support would be quite difficult. You'd really want to start from a newly-developed engine such as the new Ladybird browser that's part of the SerenityOS project.