In a real p2p coin system everyone should have one share-coin each time he is online.
Not in a Proof-of-Work system. The people that mine the more, get more coins but it is still a p2p network.
If you can share a coin through the network like an mp3 file then why you mine it? Mining is a stupid procedure. It is waste of time, money and energy.
Don't avoid the subject, you were wrong, Bitcoin is p2p.
I agree, it is a waste of energy when we have Proof-of-Stake but this is an other topic.