That's not proof of fullnode, only proof of hash power. You can mine without being a full node as long as you have a full copy of the blockchain.
http://en.wikipedia.org/wiki/Duck_testEvery running miner (not a pool-miner, but owner of full node in setgenerate=true mode) works for decentralisation and been paid fees for founding blocks.
Such miners *must* accept transactions from the network and *must* push found blocks back to a network (otherwize it is wasting energy)
That's subtly false.
Miners and full node owners are not the same. Though they both work for decentralization (as long as they don't own too much of the hash power / connectable full nodes) and the security of the network, the way they do so is completely different. See my comparison earlier. Someone can be both (as in your example - owner of full node with setgenerate=true), but someone definitely can be one without being the other.
One important distinction - miners are not required to include transactions in their block (read up on the O(1) block propagation discussion to see why this is important), so they are not required to do anything other than accept blocks from the network and push their blocks to the network. And since mining can be pooled, one copy of the blockchain can support many miners.
The problem is that mining on general devices is economically unreasonable. But we can do nothing with it.
You must understand that nobody will pay you only for your computer switched on.
I placed here a link
http://en.wikipedia.org/wiki/Tragedy_of_the_commonsFees and block reward - is a "common resource".
Running a full node is much more than just computer switched on. It requires storing the entire blockchain (~20gb right now, but will scale as the network grows - and there certainly will be a financial price for all the hard drive space that'll take up once the it's growing fast enough). It requires accepting incoming connections, and accepting and sending on transactions to anyone who needs or wants to send them (8 mbps according to the above article assuming 2000 txps, which is certainly not free).
The article I linked claims that these problems are easily solvable as long as full nodes are running on high end servers. The price of a high end server is enough that most people won't pay that price just to support the bitcoin network.
The common resource we need to protect is not fees or block reward - that's reserved for specific people (miners). The common resource we need to protect is the security of the network (you know, the thing that gives bitcoin value).