Would this make it possible to have the whole page in <noscript> tags to make it crawlable by all bots and browsable in Links, Lynx, Dillo, w3m and offByOne?
Yes, that's a good idea. You'd just need to introduce a little wrapper around the client-side rehydrate to move the contents of the <noscript> tag out of the <noscript> before calling React.render.
Then again, you don't really need <noscript> in that case. It's rendering server-side, so the page already displays correctly, you just need to make sure all your links work.