SearXNG is metasearch software. A server you run, or a public instance someone else runs, asks Google, Bing, and other sources, then shows you a combined page. Those upstream engines see the instance’s address, not yours, if the instance is doing its job.
The operator of that instance still sees your queries. A random public server is a stranger holding the postcard. Self-hosting is the point of the project: the program is open source, there is no company, and you can turn sources on and off. Public instances go offline, get overloaded, or start logging. Treat searx.be and its cousins as examples, not as a permanent home.
Use it on hardware you control. Do not paste your life into an instance you found on a forum. searxng.org
Self-hosting SearXNG is a small web app plus a list of upstream engines you can disable. Turn off the ones you do not want contacted. Rate limits and CAPTCHAs from Google are normal. A public instance that asks you to disable your ad blocker, or that suddenly adds a login, is an instance to leave.