For PyroCMS, it will be a lot easier to abandon it for the new website and keep it running for the forum only, along with PHPDocumentor setup, hence my question about content managment.
Headless CMS only hosts content, not the website. It is then served on the fly at page load through JS, from CDN. Have a look at Prismic and Storyblok for a more detailled insight.
For Mattermost, I looked and I like the Open source mentality behind it, but Discord is popular and widely used in the dev community, it may be better to encourage contributions and interest in Fuel.