If you're not happy connecting your phone to the network, then how likely is it that you would you be willing to enter your login details on a machine you don't control?
> willing to enter your login details on a machine you don't control?
Are you talking about your phone here?
What if you just can't have internet on your phone? Like if the computer is connected via Ethernet and there's no wifi network you can connect to? What if you're abroad and have no roaming? And the ultimate question about a person that the modern world can barely conceptualize - what if you have a dumbphone? Or what if your smartphone is lost or stolen or dead and you need to access some account? That last one has happened to me, and I sure am glad I know the key passwords that I need for survival.
These may seem like nitpicks, but there's probably a thousand rare scenarios like these that exist. You inevitably have to consider them when you're moving from punching in letters and numbers that you remember in the normal, low-tech way to a complex networked two-device workflow.