Ok, after a bit of effort, I managed to get a fresh generic query to produce results from a blocked source (Reddit):
https://assistant.kagi.com/share/5badc9e5-a8be-4c31-830e-f759736f09d2
curl 'https://assistant.kagi.com/api/v2/conversations' \
-X POST \
-H 'User-Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10.15; rv:155.0) Gecko/20100101 Firefox/155.0' \
-H 'Accept: */*' \
-H 'Accept-Language: en-CA,en-US;q=0.9,en;q=0.8' \
-H 'Accept-Encoding: gzip, deflate, br, zstd' \
-H 'Referer: https://assistant.kagi.com/' \
-H 'Content-Type: application/json' \
-H 'Origin: https://assistant.kagi.com' \
-H 'Alt-Used: assistant.kagi.com' \
-H 'Connection: keep-alive' \
-H 'Cookie: turnstile-session= \
-H 'Sec-Fetch-Dest: empty' \
-H 'Sec-Fetch-Mode: cors' \
-H 'Sec-Fetch-Site: same-origin' \
-H 'Priority: u=0' \
-H 'TE: trailers' \
--data-raw $'{"content":"So what are people sayin\' about Kagi?","model":"ki_quick","enable_search":true,"lens_id":null,"personalization":true}'

So it seems like here it explicitly looks at Reddit to get a response to my query, which means it's not referencing the personalizations/filtering before it goes out to gather information perhaps?