It's not that the router is broken, you have to remember that home base routers are just basic router with four or more switch ports build into it. So your LAN may appear to be working, file sharing is working..because the packets never leaves to the default gateway where it is handle by the routing feature of the router. The only time data leaves the network is to another network like the Internet, so the packets go through the default gateway where the router determine where to route it to which interface. This usually indicates to me, that some feature on the router may be enable like Stateful Packet Inspection and other firewall features that can interfere with the communication process.
It seems like you have connectivity on the lower layers when you can ping the websites. So your connection seems fine, but not at the application layer. Possibly a firewall or proxy issue either on the computers or the router. But I'm leaning more towards the router. So your best bet is to find out by connecting a computer directly to the modem.