Maybe this is a PEBKAC but here's the behaviour I'm seeing:
Kagi is set to "New windows open with Start Page" and "New tabs open with Start Page". If I open a new window with Command-N, a window correctly opens to the start page. If I then try and close this with Command-W, it will not close and I need to do Command-Shift-W instead. However, if I open any other site from the start page, then I can close it (and the window) with Command-W.
So: Start page won't let me close it with Command-W if it's the only open page. I hope that makes sense.
I expect to be able to close the Start Page with Command-W when it's the only open page, as I can do so with any other open page.
Version 0.99.133-beta (WebKit 621.1.2.111.4)
Sequoia (15)