This isn't really a "Linux" problem. Calling it a Linux problem implies all distros do the same thing out of the box because it's a part of the core system. Systemd has a file, /etc/systemd/resolved.conf which has one line DNS= that you can add the servers you want. It's as simple as that. If you're using Dnsmasq for DNS instead, you'd edit the Dnsmasq file. If you're not using my of those (i.e. you removed systemd-resolved, Dnsmasq, etc) then you can just edit the /etc/reeolv.conf directly without worry of it being overwritten.

While many distros come with systemd out of the box, not all of them do. For example, I use Gentoo with rc and after editing my resolv.conf, never had to worry about it again unless I decided to install a custom DNS software on it later.

I read many replies to your post as "DNS software shouldn't be allowed to change DNS settings" for the most part, and that doesn't quite make sense to me. If it's a problem, remove said software. Browsers are definitely annoying in the DNS front, I won't disagree with that. Fortunately, they allow you to turn that off though.