MBP works fine but not visible on network
So, Broken MBP can ping anywhere, including other machines in your network, but can it ping itself?  (You have to read that with the "Will it Blend?" sound)

The rest of the machines can ping each other, but not the Broken MBP. 

Am I correct?

Does this only happen via Wi-Fi? Have you tried using a Cable?

What happens if you turn ON the firewall and make sure that in "Advanced" the machine is not in Stealth Mode:



(Security -> Firewall -> Advanced)