The browsers could buffer the text in text fields until the field loses focus (clicked another field or submit) or hit enter. It would break search-as-you-type functionality, but not the biggest loss in the world.
There's always the option of typing your stuff in another program entirely and copy/pasting. Sure, not everyone would be willing to do that, but it's really not that inconvenient if you know your keyboard shortcuts (ctrl+a, ctrl+c, alt+tab, ctrl+v in that order, easy for a competent computer user). And socially anxious people may be more comfortable with that overall.
There are quite a few browser extensions that make this easier. The best is Pentadactyl, a complete UI overhaul for vi bindings everywhere that lets you hit Control + i to open any text field in any (blocking) external editor (gvim -f by default), so you can type out your message and then type ZZ to send all the text to the field at once. However, it’s only for XUL browsers. I think there are also WebExtensions that offer similar external editor support, but when I use browsers limited to them I use wasavi, which pops up a javascript-powered vi clone inside the browser (but still only sends the text down to the area when you save). IIRC SurfingKeys also has a similar thing, with a codemirror-based editor that’s IME more limited than wasavi. Of course none of these work on exceptionally complicated things like Google Docs, but they often work on simpler WYSIWYG rich text areas, and wasavi and maybe SurfingKeys even support converting the HTML within them to Markdown while editing, and back when you save.
Copy+paste won't trigger keyboard events to my knowledge. So if the page was just simulating a text input field, paste wouldn't work. (Which would cause issues e.g. pasting account numbers to be fair.)
Browsers already have good UX reasons to do something like this. They should offer version history on textareas, for example, so you don't need to ever worry about losing state.
But they don't. So I build it (draft system) for my users by saving what they type as they type it, and they are very thankful.
If you would kill my ability to build that feature, then you better replace it with a solution.