I don't think turning off DNS servers is 1) actually possible (people would start running mirrors + other countries continue to run) and 2) would disable the internet into uselessness (anything IP based still works).
It doesn't have to be 100% effective, just as with the Great Firewall. Shutting down DNS would shut down most existing apps until they're updated (and also the app stores that are normally used to update them), and preclude the majority from accessing anything online.
Despite the centralization efforts of big tech, Internet is more resilient than you think it is.
> would shut down most existing apps until they're updated
Most apps are using DNS client APIs provided by their operating systems.
> and also the app stores that are normally used to update them
They too.
As for the OSes, they don’t normally have hardcoded DNS servers either. Instead, they rely on DHCP to get DNS servers from the routers. There’re many levels of caching involved: operating systems have a DNS cache, home routers have a caching DNS servers, ISPs usually running their own caching DNS servers for their clients, and so on.
Updating any single one of them going to fix the issue for all of these end-user apps. ISPs probably gonna be there first, it’s their job to support their network infrastructure 24/7.
If it gets to the point where DNS is shut down by government fiat, ISPs would be unlikely to push back, because that would just make them a target for more, shall we say, physical activities.
The servers that are run by the ISPs, which is what is used by the vast majority of end users in practice (usually indirectly via their router, as you've mentioned).