I would never use my own "private" instance precisely because of privacy reasons. If all search requests from an instance are by you then you haven't really changed anything. Whereas if you use a public instance you're hidden in the crowd from the perspective of the search engines that the instance relies on. You should use it via Tor of course, like everything else that you want to keep anonymous.
For similar reasons avoid using only one instance "provider" (like snopyta) for everything (invidious, nitter, bibliogram...), because that means snopyta has access to an aggregate of your browsing, similar to an ISP or a VPN. It's better to use official instances instead because they're usually isolated from each other.
>I am still trusting a 3 party with my data
You should never trust anybody in the first place, and instead technically reduce how much damage can potentially be done to you. Trust is unreliable. Use Tor and compartmentalize your browsing.
All of these search proxies have gone to shit in the past few years. I think the search engines feed them computationally "cheap" results in lieu of simply blocking them, which is what they did at first. It's probably a better strategy for Google to not block anything and instead make it shit. Won't cause controversy and people will still flock back to you.