Well, he wants to access his limited security account from a trusted machine that belongs to someone else, which is not quite the same thing. You're looking at it from a black box security point of view, but in practice he knows the parties involved, he knows what the security set up is, and he knows the degree of control he has over the situation, and given those inputs he's making a perfectly reasonable decision to enter his password into the box involved.
And yes, there are no good password managers, because they don't handle that situation, and that situation is entirely reasonable and normal. You can tut tut, and say that from your point of view there's no difference between entering your Amazon password and enabling one-click purchases on a laptop left in an airport with a yellow sticky that says "FREE INTERWEBS ACCESS HERE HONEST!!", and entering your Netflix password on a branded Roku box owned by a friend, watching a movie with him, and then logging out afterwards to prevent any accidents. But actually, no, they're not the same situation.
One of the downsides of computing is that when it comes to "security" or a whole host of other processes, some people design "solutions" for a specific set of cases and then decide that those cases are the only ones in existance, because it's easier to pretend they are than to actually produce something that solves the other problems. From anti-spam "solutions" to
password portals, we're still trying to grasp the fact that these solutions don't work. At the end of the day, developers have seen the "bug between chair and keyboard" make so many problems, they forget that it's still the case the person they're hobbling is the person that's trying to get shit done. Design an impractical security system, and you're making things less secure, because you've just wasted time - yours and the user's - building something that'll never get used.
This is why I hate password managers as a security device. As a convenience, so I don't have to type "squiggleslashamazonAAA" every time I log into Amazon? Sure. As a security system, so nobody - not even me - will ever figure out my password is "09F911029D74E35BD84156C5635688C0"? Fuck no.