You can ask claude to write you a 10 lines userscript that integrates with google/ddg/kagi/whatever is your search engine, and hides the websites from the result page.
Generally, you shouldn't do server side what you can do on client side
They do have anonimised logins for this though: https://blog.kagi.com/kagi-privacy-pass which is a pretty good mitigation IMO. As it's a paid service of course proving you paid is a must.
And as for control, I can't agree there. Kagi offers more control than any other search engine through its lenses and the ability to influence the ranking of search results from specific sites.
I don't use their service at the moment, I'm pretty ok with my self hosted SearXNG and I like being able to customise the look and feel there too. But Kagi is excellent as search engines go.
I'm doing it for having a clean adfree experience, having multi-engine searches and having control over which engine and features it uses. And it also helps really well against search engine enshittification by raising search results higher when they're received from multiple engines so you have less of the clickbait crap that search engines promote these days.
It has some amazing features where it can search much more specific sites if you search for things like books, movies etc instead of treating everything like a general search. And everything can be tailored and tweaked.