I got a new router (Asus) because my old one is getting flaky and requiring frequent reboots. My problem is, I have one desktop that connects via wifi and I can't get it to see the SSIDs of the new ...

You can do tons of stuff with a computer that most hardware NAT-routers cannot do, like having firewall software run on it, provide services (print (I know the SMC routers also have this), file, proxy ...