This will only truly make sense to another screen reader user, but here goes.
NVDA, the screen reader I'm using, has two modes. When it lets me type, as in what I'm doing now, it is in focus mode. In browse mode, arrows and special hotkeys navigate the webpage and typing is impossible. Normally, it enters focus mode when focus lands on a control that expects typing or arrow navigation, like textarea, input, and so on. When focus leaves the control, or the page loads, this mode is automatically exited and NVDA returns to browse mode.
With Kagi, NVDA remains in focus mode, which is odd.
- Open kagi.com while running NVDA. You are placed on the search field, and forms mode is entered automatically.
- Perform a search by typing a string and pressing enter.
- The page loads, but NVDA remains in focus mode. It should return to browse mode so you can start exploring your results.
While this may be an NVDA problem, I haven't seen it happen on any other webpage. Also, though I'm in focus mode, I can't type.
When the search page loads, NVDA should return to browse mode. Perhaps you could automatically set focus to the first result in the list? That should do the trick.