You do realize the amount of steps your leaving out? Either you work for Google or simply anti-Apple.
How did your wife activate google asst on the iPhone?
How often does the Chromecast break and need to be reconnected?
What about security, your wife doesn't care that the kid next door can hack Chromecast and access what you streaming very easily?
How about the FACT that Google actually allow their employees to view your photo AND know who took those photos, when and where?!
Then that lead us to the FACT that Google also tracks her and your polictical views, who you support, what causes you support, health issues you've researched, etc etc etc...
All that aside, no one has mentioned the important FACT about iOS 11 and Siri sharing "learning" via others interactions randomly... i.e. Your at a coffee shop and all the iPhones will randomly connect and learn from each other! Something that Google does NOT have the numbers on or can with Google Asst. on the iPhone.