Wouldn't it be much more fair to give miners only the TX-fees, and pay a reward to some random stranger (transaction) in that block? That way the initial distribution of coins is garantueed to be fair for everyone, since the coins are going to the actual users of the network (which includes also 'the poor'), instead of the big guys.
Block rewards were never intended to be a lottery. Amongst other reasons, it is an incentive for miners to do the work required to secure the network.
And if the 'random reward' was sent to, what-- random addresses? Everyone would be mining addresses.
If the 'random reward' went to random nodes-- those with the resources would have thousands of virtualized nodes or giant RaspberryPI farms..