So the router is handing itself out as the DNS server (which is fine in the right circumstances)?
You probably want to check the output from: resolvectl status
to see what DNS servers are available to the computer and which DNS server is being used by it at the current time. I think this would be more definitive than /etc/resolv.conf
On my system /etc/resolv.conf is a symbolic link (into /run/...) and those target files are all generated dynamically. The point is that each interface contributes one or more DNS servers and as each interface comes up and down the total list of DNS servers changes dynamically (both because of the interface coming up and down and because often the DNS servers for an interface are obtained dynamically via DHCP). That isn’t really compatible with the old static way of configuring an actual file, not a symbolic link, as /etc/resolv.conf
So you should confirm whether /etc/resolv.conf is an actual file or a symbolic link. And, yeah, if the file (either way) says “Generated file. Do not edit.”, that’s probably good advice. 
You need to fix the underlying problem.
So my system is using the systemd-resolved service for DNS handling. If yours is too then you might want to check output from: systemctl status systemd-resolved
A lot of software won’t even use /etc/resolv.conf to determine how to resolve hostnames but some legacy software might and that’s why the file still exists.