Some further searching revealed the following thread, where others are reporting that the status never changes (Chromecast Beta Dead?). For a workaround, someone suggested creating a virtual switch in HE, exposing it to Google Home, and creating a GH automation that sets the switch based on whether the Chromecast is playing. However, the new Google Home automations editor currently says "Support coming soon" when I try to use the Chromecast as a trigger.
For now, I created two rules based on the "Media Source" state of the Chromecast. This state appears to update immediately once I begin casting, and eventually back to "none" after a significant (but acceptable) delay. So far, the rules appear to be working, so I'll test this for a little while. And make a backup of the HE config 
