if it is systemd-resolvd, then privacy oriented solution would be to not use google, cloudflare or even the non profits. It would be to just have list of 13 DNS

if it is systemd-resolvd, then privacy oriented solution would be to not use google, cloudflare or even the non profits. It would be to just have list of 13 DNS root servers and do recursive name resolutions (i.e. what caching resolvers were meant to do). But the absolute correct behavior would be a hard failure, because if you have IP properly assigned and routing setup but no DNS, then you either purposefully set it up that way and don't want DNS to work, or you have misconfigured network setup which you should be aware about it, without having to troubleshoot why some machines don't have working Internet while others seem perfectly fine.