Are there any network/router level options for search privacy? For example, is there a way to force google to its &udm=14 self? Or even redirect queries from one search engine to another (like Google → DuckDuckGo)? Possibly something better I’m not even considering?

Update: I haven't been able to find a solution to this. It's possible to do using Google Chrome (and a few other browsers) on individual devices. If you use Safari (like I do), you'll need to use the StopTheMadness Pro app and follow these directions. Even if you don't want that specific feature, I highly recommend StopTheMadness Pro. I've been using it since the day it launched.