Personally I would just like the basic keyword functionality before we start thinking about GPT powered anything. I don't even care about "alerts" as such. If I could cut out the middle step and be able to create an RSS feed based on any search query (particularly on the news tab) that would be amazing.
In regards to your earlier questions.
Authentication - how do we authenticate the user and prevent abuse
Probably require the API key as a parameter or generate a unique UUID for the feed when it is initially copied by a user which achieves a similar thing.
Cost management - if results get generated on every pull of the feed it could be a lot every day (and this will all cost the same as regular searches)
Personally I'd be happy for the results to be cached on the kagi server with some kind of regeneration limit to avoid constant regeneration with overhappy RSS clients. Basically something like serve the cached version 1-30 mins after generation then regenerate on a request 31+ minutes after the last regeneration.
Interface - where do we put the rss feed icon in results?
I don't have much opinion on this one but probably the feed icon next to the search bar would be fine. Also advertising it in the standard paths so RSS aware browsers/extensions identify it automatically.