Does it have to be Chrome, or just e.g. Chromium is fine?

There’s stuff like Searxng or whoogle, but these aren’t “real” search engines, merely “search aggregators” - they relay requests to a bunch of actual search engines, like bing or google, and aggregate the results. That’s why they don’t require tons of compute and scraping, and also why they often fail to work (since the search engines in question don’t like or allow this). I believe it’s not feasible to run a “real” search engine alone or even as a small group of people - according to this comment you need a powerful server with terabytes* of drive, hundreds of gigabytes of RAM and a lot of compute - and all of this will just let you crawl some top domains, nowhere near a good chunk of the internet.

*which sounds low actually, I would have expected more for this

Lutris doesn’t, but you can do firejail --net=none lutris

OBS supports capturing audio from a specific application, so if you set that up you don’t need to be careful with foreign sounds.