What happens is that you can’t do it. Maybe that’s a feature? Passkeys prevent you from logging in using a dodgy hotel computer. It’s protecting you from yourself. I’m not sure that’s appropriate for every account, but there are probably some accounts where it makes sense.
But I expect that businesses will probably like it better than consumers. They probably don’t want their employees logging in using dodgy hotel computers.
I will agree a hotel computer is dodgy. I did try and give it a once over in the OS, and checked for any obvious hardware keyloggers before doing anything. I also changed my password as soon as I was done. The alternative, if I remember correctly, was not making it home. With a password I can evaluate a situation and take a calculated risk to solve a problem. If I had passkeys, I would have been out of luck. Maybe I could have spent hours with some customer service agents to try and figure it out, hopefully before the flight left. If a passkey left someone stranded in another country, any idealistic views they might have would quickly be replaced by frustration and anger.
The other 2 examples I gave were not random public computers. They were either in my control or the control of trusted family members. I still want solutions to those situations that passkeys can’t answer (as far as I know).
There are similar issues with 2FA. I was traveling a few years back and broke my phone. It’s the only time I’ve ever broken a phone. All the info for my flight and my tickets were on the phone. I was able to get to an Apple Store and get a replacement. When I went to set it up I got a 2FA prompt (it was enabled for me without me opting in sometime earlier). The only reason I was actually able to set it up was because I brought an iPad with me, which was just dumb luck. I often only have my phone. On my most recent trip I created a recovery key, wrote it down, and put it in a money belt I wore everyday. I’m really not sure what other option I’d have to recover if my devices were broken/lost/stolen. Of course having the key on me carries its own risk. Yes it was hidden and on my person, but I also felt the need to add a bit of randomness to it, incase someone did somehow get it, somehow figured out what the paper with a bunch of seemingly random letters was, and tried to use it. But this isn’t a normal thing people do, they’re just going to be screwed if something happens. When security starts locking out the owner because it’s too unclear, too complex, or too device restrictive, it can hurt more than it helps.
I understand the issues with passwords and why people want to get rid of them, but this feels like a happy path solution that doesn’t account for edge cases, which is a problem. There will always be edge cases and they can’t be ignored for something as foundational as authentication.
I'm wary of not having backups, too. For passkeys to be more than a niche solution, I think the answer for travellers will be not having just your phone, in case it breaks. This is a hardware-based solution so there needs to be extra hardware.
I have a Yubikey on my keychain and usually have a tablet as well, but it will need to be something more common.
To speculate, credit cards have RFID chips in them now, so maybe there is a possibility to identify yourself well enough to buy a phone and restore backups?
Meanwhile, if you use Google and Apple password managers, they do have systems to get your password manager back on a new device. For Apple, it seems you need to remember your AppleID password and for Google, the pattern you use to unlock your phone.
But I expect that businesses will probably like it better than consumers. They probably don’t want their employees logging in using dodgy hotel computers.