In reply to Unai.

You can also check the project I'm building: https://getpop.org (sorry for the self-promotion!). It basically wraps WordPress in a way that allows it to behave like a headless CMS. The installation scripts are not ready yet (I need a few months more of work), but the concept works like a charm, allowing to do absolutely everything that you can already do in WordPress (including authentication) through standard HTTP requests, and getting a JSON response back (it is not REST though). And it's pure PHP, Handlebars and vanilla Javascript, so other than Handlebars (just for markup, and very easy to use), it's pretty much the same technology stack as WP (the open source repo is at https://github.com/leoloso/PoP). Just check it out, I hope you like it :)