Sounds like a solid plan.
Unless you already have a RPi available for use, I would consider basing around x86 hardware. Nothing wrong with the Raspberry Pi’s, it’s just if you need to buy hardware, there’s much more powerful computers available for similar price points. Won’t be as simple or straightforward nor as energy efficient, but if you start adding additional services, you’ll be glad for the extra RAM and processor performance that is available.
I use SearcXNG. It can use Google on the backend, but in default configuration it uses several different backends. It can also use paid backends if you set it up. Basically if it has a search api, SearXNG can hook into it with the right configuration.