When I went through the process the company was very small and Vlad personally reviewed applicants, and he used take home projects like this one. It seems like the company has grown larger so now he isn't looking at them himself anymore, but the essence seems the same. If you ever talk to him you'll find that he's basically the Hacker News commenter stereotype and his interview is really a vibe check to see if you seem cool. To him, submitting a long design doc where you go "I am going to use Galactor. I will make sure the project is florp-ready. I will fleem." is the exact opposite of cool. When the assignment says "I want a terminal-inspired email client" the project that passes with flying colors has keyboard shortcuts for everything and comes with a paragraph explaining how it never takes more than 2ms to draw a frame. I suspect the interviewer thinks you are too "enterprise-brained" to work at Kagi.
Now, is this actually a good way to screen people? A bunch of other people here have opinions on it, so I won't rehash it too much. There are obvious problems with it. Vlad basically wants people who think like him. This kind of interview has a lot of problems that have been discussed and studied at length. But if you understand that and are privileged enough to go through that process, the task will actually make sense to you. I think Kagi could do a better job at communicating this. It's unfortunate that you had to waste a bunch of your time to get to this point but I hope you can find something else that matches your working style better.