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Oh, that's how Kagi Search extension works in Chrome. Thanks for clarification!
I'd be down to switch to the Kagi Summarizer extension, but I've had a couple of problems:
I obtained the API token from my Kagi settings (https://kagi.com/settings?p=api&generate=1), entered it in the designated field, and clicked Save token a few times and restarted Cchrome a few times.

However, it won't use the token(?) for any paid engine(Agnes/Daphne/Muriel). I believe Agnes and Daphne should be free for Kagi users, with Muriel as a paid enterprise model. However, all models are showing the message.

And the free version does work for some links but doesn't work on some HN links like https://news.ycombinator.com/item?id=43137616. It used to work, but now it doesn't. Let's address this separately as it may be unrelated.

I checked my Kagi API settings (https://kagi.com/settings?p=api) and saw that the credentials hadn't been used when I used Agnes/Daphne/Muriel engines. It seems the widget isn't recognizing the token.

Could you advise whether Agnes or Daphne would replicate Kagi Search? Once this is settled and the widget token is fixed, the problem will be solved.
Thank you!