1. Why would a miner do this and not include any transaction missing the fees?
To reduce the size of the block, thereby reducing transmission time and the chance it will be orphaned, which would lose not only the transaction fees but also the block subsidy. As the block subsidy decreases, so does incentive to drop fee-paying transactions, so this will not be an issue in the long term. In the short term, raising transaction fees is an option if too many miners start doing this.
2. If everybody started doing this bitcoin will be mined but no transactions will occur, is it a flaw?
It's a flaw in the miner's business plan once people start paying greater transaction fees that the miner is missing out on. Competition will ensure that only miners who actually include transactions (and take the fees) will remain profitable.