You can, and should, make any API endpoint CSRF exempt.
Django offers the csrf_exempt
decorator for exactly this, see https://docs.djangoproject.com/en/dev/ref/contrib/csrf/#django.views.decorators.csrf.csrf_exempt.
Also CSRF is intended to prevent unintended actions being performed via GET request forgeries. It is not intended to make it impossible for an automated system to submit forms, there are captchas for that.
As for what prevents you from using AJAX to grab the whole site and extract the token is something called the Same-Origin Policy. This is implemented by the browser and prevents any AJAX call from returning data when the target of the AJAX call is a different domain without the correct headers set. (I'm not entirely sure what sandboxing is applied to chrome extensions concerning this). As such it will, or at least should, fail to get data via AJAX for normal websites, e.g. a profile page. If you want to interact with third party websites you should look into whether or not they offer an API.