The internet does not work on top of DNS. DNS works on top of the internet. Nothing about IPv4 was circumvented here, and IPv6 is even built to not need/use DHCP at all.

There is no mandate or expectation that all DHCP clients always use the advertised DNS settings. If there was than alternate DNS services like 1.1.1.1 or 8.8.8.8 wouldn't even exist in the first place, as your ISP would be in complete control of your DNS settings unless you splurged for a static IP.

The cast client decided to use its own DNS settings as many clients do - you can override the DHCP settings on just about any general purpose OS, for example. Even though they are DHCP clients. Do you call Linux allowing you to specify a DNS server as "circumventing how the internet works at a pretty basic level", too?