I have a similar setup on my laptop, a docker searxng (well podman rootless, but near enough) locked into a gluetun instance. Works fine, simple to set up, sucks less than any individual search engine and is usefully configurable, but I’m on linux, I expect there’s more pain for windows (linux might be a use for the spare computer…). It’s not resource intensive. Gluetun let’s you expose a local port for searxng and you just point your browser at https://192.168.x.x:8192 or whatever, no need to worry about exit IP. Gluetun is well used and has a focus on avoiding leaks, plenty of eyes on the code, I’ve never had any problems.
You can wireguard (or tailscale or whatever) into your home network and use it on your phone too. Spin up a pihole for adblock while you’re at it. I say go for it…
Perhaps a poisoning the well attack on AI scrapers?