If routers worked that way, people would go crazy resetting their routers. Think about a company with 1,000 computers, each of them may be rebooting in any given minute. They would have pretty much 0 internet connections if they had to restart the router each time a PC rebooted.
The router runs on it's own system, the only time it's linked to the PC is though the network connection which can be broken and setup at the computer at any time without the need to reboot the router.