In particular, a modern high-end router has a CPU (which might be an x86 of some kind) which handles administrative and auxiliary tasks (NTP, ssh, DHCP reflecti

In particular, a modern high-end router has a CPU (which might be an x86 of some kind) which handles administrative and auxiliary tasks (NTP, ssh, DHCP reflection, whatever you want) while making one or more dedicated ASICs handle packets. When you manage the router, you're actually talking to the supervisory CPU.