If you switch from Chrome, closing Chrome (including turning off the computer) will stop the broadcast, no matter if it drops the tab or casting from the application / web page ready for delivery.
If you want to host your web page, you need to write an application that uses a custom receiver. A custom recipient can load web pages. If he just launched your only web page, then just get it as the recipient. If you want to use it as a browser and use different web pages, you can do something like your sender passing the URL to the recipient (maybe use an iframe to load the web page on the receiver). For the latter, tab casting is probably much better.
I suggest you take a look at Google Cast samples on the Github page. HelloText-Chrome is probably a good place to start to familiarize yourself with Cast.