You don't need your email set up on someone else's PC in order to log into a service which uses passwordless login. You only need access to your email from some device - not necessarily the device which you want to log in with.
As I understood the service, it does not require that you click the link on the same machine that you login from. In fact, the screen shots sort of assume that you open the mail on your phone even if you login from your desktop.
In this day and age, you should not login to anything from someone else's PC.
There are a lot of occupations where you don't have your own computer, and share one with many other people.
For many jobs, the value is not in the person, but in the position, so the position has a single computer for a function that multiple people fill. Especially if you work for a company that operates 24/7.
For example, each person person operating a control computer at a recycling station does not have his own computer. There is a computer for each position, and who mans that position will change from day to day.
Freight dispatchers, retail sales, customer service, airline operations, and many broadcast positions are in the same boat.
Imagine how big an airline ticketing counter would have to be if "In this day and age, you should not login to anything from someone else's PC" was reality.
That's a silly interpretation of my point. Sure, log into work resources - including your work email - with your work computer. Email token auth works fine for your work context.
If you're mixing work resources and personal email (or vice-versa), you have a problem. You should never hand off control of your intimate personal life to the security (or lack thereof) of your employer. And vice versa; your employer should not trust putting their intellectual property on questionably secured personal computing devices.
In that scenario, do you have individual accounts on the computer or are you all using a single login and sharing it? The first scenario is fairly secure, the latter is a nightmare.
In that scenario, do you have individual accounts on the computer or are you all using a single login and sharing it?
I've never worked for an airline, so I can't say what happens in that specific scenario.
However, when I worked for a chemical company, printing out the hazardous materials labels for sample drums that were shipped all over the country, the computer had a single common login for all users.