Congrats!
The data is fetched through the launcher/brave browser, which itself exposes the content via an express nodeJS server (the headers you can see). It is “running” on your local system. Technically you are already “running your own”.
And that is entirely possible but requires a different architecture of the app itself. As an example we have (/had) community prototypes for a blog engine, a decentralised forum and a drink collection app – all serving “non static content” right from the network itself, needing nothing else than the browser to work.
(hint, I am also working on another demo that runs in the browser exclusively using that new lower interface @anon40790172 refers to)