There is so much Wrong information here that goes against everything I thought.
Maybe what you thought was incorrect not the information presented. Google "
block withholding attack". It has been know and discussed pretty much as long as there has been pooled mining.
3. Client solves the new "problem" (Important; Client dont know if their solution works on "complicated problem" as client dont have that)
This is incorrect. A miner instantly knows if he solved a block or not. However that solution is tied to that specific attempted block (with predetermined coinbase or "reward" address). A miner can't profit from this information. He can either submit it or not submit it. Not submitting it is the block withholding attack. This will reduce the gross revenue produced by the pool and will equally affect all miners in the pool (including the attacker) and the pool operator.
There are ways to make the block withholding attack impossible however they require a hard fork in the protocol (a change in the format of the block header).