So, the gif is misleading because what it actually shows is opening up the native app using a custom protocol (itchio://whatever).
But, in fact, the app is native and the primary use case is to download+install native games.
We plan to support web games (HTML5 + Flash) as well, but it's a bonus/nice-to-have feature to be able to play web games offline / preload them for a let's play or whatever.