The Chromecast goes to its own HDMI port on the TV, so it's not connected to or through the TiVo at all. There's a Google Cast extension that you add to the Chrome browser and this handles the transfer of content from the browser to the connected device. When the extension is running there's an icon in YouTube (or other compatible webpage) that you click on to send the content to your TV, like this:
A list of compatible devices appears when you click the icon. Until recently there was only one device on the list, my Chromecast. Now, in YouTube, the list shows Chromecast and the TiVo Premiere.