How could there be a page 2 of the archives if you’re not allowing archives?
This is also why it’s working that you can have an actual page at that url and add a loop…you’re adding that loop.
If you change your url “announcements” to something else you’ll still see the first page of your cpt because presumably you’ve added your own custom loop.
You should not be rewriting your cpt permalinks to a page you have or create a page with the same url as your cpt permalink. It leads to confusion as you see in this case.
If you turn on the archive of this cpt, refresh your permalinks, you’ll see that your content is no longer available.
Id suggest changing the slug of the page url to something else. That’s the way WordPress is intended to be used.