I will likely end up using this in personal capacity, and would also appreciate if the UX of using Google Authenticator were more similar to this, rather than requiring me to screenscrape my phone with my eyeball and then type information into another device (or, more painfully, another window on the same phone).
The easiest way to do that probably results in a callback to Big Daddy G every time I access anything sensitive and I'm cool with that.
With Apple's devices, you can copy-paste your 2nd factor token between devices, which makes things much more convenient. Although cloud copy-paste is still significantly slower than tapping "Allow" in a notification, it's lower friction than manually typing those numbers.
Although the security of cloud copy-paste I haven't investigated...
I've not checked how it works, but I don't think Universal Clipboard pushes the data that's being copy-pasted via the cloud. I've certainly used it to copy Wifi access codes from my phone to laptop when neither device had internet access, so I think it's implemented using peer-to-peer Bluetooth/Wifi.
The easiest way to do that probably results in a callback to Big Daddy G every time I access anything sensitive and I'm cool with that.