Interesting. I have always used their web app (even on mobile, i just use their pwa instead of the native app since the native app is missing obvious features), and I haven’t had any issues, but I can definitely understand the frustration if you want to use anything else. OP, keep that in mind if you’re thinking about Proton!
I primarily use DDG, but the vast majority of people I talk to (including other tech savvy people) use Google. I feel like “search” is too generic (search where?), but “search the internet” is weird. And saying “Bing it” or “Duck it” or whatever just sounds overly contrarian. But if I say “Google it”, people know exactly what I mean.
So yes, I will “Google where to buy some bandaids” by searching DDG for adhesive bandages.