Using the Kagi assistant, with the NVDA screen reader on Windows, there are some unlabeled buttons and the model chooser is set as a button but should be an expandable button or dropdown.
Starting from the top of the page, the checkbox after the "Control Center" link does not present its label when using the up and down arrow, but does present its label when I use the Tab key to navigate to it. This will be an issue for those who prefer to read the page a line at a time instead of tabbing through elements. This could be an issue with Aria use, having a description but not a label.
After the button that says "private by default", my message is shown that I sent to the assistant. However, in longer conversations, I cannot tell who sent which message, me or the AI. It would be nice if "you" and "assistant" were put as headings before each message. Alternatively, if headings would mess with the visual flow, landmarks can be used instead.
On the references heading, the references are still shown to the screen reader, even though the button is collapsed.
The button after the references, before the "enter prompt" text field, is unlabeled. This may be the language menu, as AI describes it as having "enu" on it. It could also be a menu button.
The next button is a bit like the checkbox above. When using the arrow keys to get to it, it is blank, but when tabbing, it says "no file chosen". This may be a browser default description, though.
The next two buttons are unlabeled and do not have descriptions. AI says the first is the "microphone" button, and the next is the play button.
The next button has the currently selected model, Claude 3.5 sonnet. When I press Enter on the model selector, I'm placed into a search field. I suppose it's to search for a model to use. Pressing Tab, though, does not let me exit the search field. Using heading navigation, I can find all the models. However, the button to open that dialog should be labeled like "Model chooser: Claude 3.5 Sonnet."
After that is the web access checkbox, with its label missing from the element itself, but showing up below the element in text.
I hope all this helps a bit.
All buttons and other controls should be clearly labeled, and the conversations separated into clearly descernable and navigatable structures.