My laptop doesn't have a webcam, and even if it did, I wouldn't permit my web browser address to it unless I really needed to. A website that asks me to access the webcam just to log in isn't going to appeal to me.

I've no idea how typical I am in this regard, though.

Could you turn the idea on its head? Could the website display the QR code and the phone scan it, as per WhatsApp and many others? Or perhaps the phone just displays the note ubiquitous "random raccoon" type wordset for the user to enter into the website?

I'm probably missing the point of your idea.