Sure, but if you are considering a malicious party in the kagi case, your steps don't help. What you propose can totally work if you are considering good faith parties.
In other words: assume you use searXNG. If you now want to consider a malicious party running an instance, what guarantees do you have? The source code is useless, as the instance owner could have modified it. I don't see a privacy policy for example on https://searxng.site/searxng/ and I don't see any infrastructure audit that confirms they are running an unmodified version of the code, which - let's assume - has been verified to respect your privacy.
How do you trust them?
I am curious, what do you use as your search engine?