IP and IP addresses are still very important because that is how information gets routed on the internet. Without IP addresses, nothing could travel over the internet at all. The missing piece that makes IP blocking not work is that it is possible to set up what are called Virtual Private Networks or VPNs, which act like real networks even though they are really mapped onto the internet itself. I could sign up with a VPN service, and any communication I did would go to that service, which would act as a gateway. If I was interacting with Google through a VPN, my communications would go to the VPN servers and then get rerouted to Google using a different IP address than my own. Any responses from Google would go back to the VPN service, which would then route it to me with my real IP address.
As you can see, the IP addresses are still used -- and necessary -- but Google wouldn't ever see my real IP address. I could then switch at any time to a different VPN and appear to be accessing Google from a completely different IP address, all from the same computer.