Assuming you control your device (non-corporate non-government users) and know how to configure your DNS server (elite users only) and wanted to do this extra step (privacy-conscious people?), this will break for hotels, airports, cafes/restaurants, corporate/government networks, anyone with an ISP that intercepts public DNS requests, and various home routers. Not to mention making surfing the web much slower and less reliable, since we distribute name servers specifically to rely on caching to lower latency and increase availability in case of various network failures.
But otherwise, sure, that works.
If this isn't the case, then I know that the device operates not in my interest, but in that of whoever controls it. So, I probably won't be entering any sensitive information into it.
Good point! I didn't know that. Perhaps that's what you're running into, since Android 5.0 seems to be working fine for me? For a long time we all just used whatever DNS server we got from DHCP, but I think the time for that has passed. At this point we should choose our own recursive DNS provider, and reconsider using networks that don't allow that.