<<<< However, the client computer cannot connect to the Internet. It doesn't appear to be registering the fact that ICS is enabled on the host computer. >>>>
ICS *ISN'T* enabled on the host computer. At the top of the IPCONFIG /ALL details note that it says:
IP Routing Enabled. . . . . . . . : No
which means ICS hasn't been enabled on the Internet connection. You also have TCP/IPv6 enabled and this should be disabled.
Other than that, you should go with the help in your other thread to connect all computers to the router. If you are connecting with PPPoE to the service provider, there is only *one* PPP connection to the service if you use a router, the "client" computers connected to the router don't use Point-to-Point Protocol (PPP) to connect.