Homey can cast a web page to Chromecast;
Homey can run a web server (Micro Web Server app);
With iCalCalendar app, Homey has appointment driven automation possibilities;
Homey has to be able to turn your tv on/off (using an app, or using the build-in IR remotes);
Here’s an example I made a while ago (without calendar events, but that’s just making the right flows for your usecase).