In your proposal, you do the opposite: you print new bitcoin bills and distribute them to the stake holders. But this just devalues the money, and assuming all else remains constant, price levels across the economy would tend to rise. In other words, stake holders may have 10% more money but everything costs 10% more.
As a result of writing the whitepaper and listening to constructive comments, e.g. yours, about the proposed Bitcoin system, I am persuaded that paying dividends to stakeholders is a lower priority than investing bitcoin block creation rewards into a network capable of processing all the world's transactions.
This system needs a way to prevent an attacker from establishing numerous puppet nodes, who by acting in concert, could outvote the correct nodes and allow misbehavior. Proof-of-stake provides a means to limit the ability of an attacker to outvote correct nodes. The assumption is that correct nodes will offer sufficient stake in a hot wallet in return for a certain amount of dividends.
The reward allocation policy is most certainly going to be contentious, and increasingly so as the technical barriers are overcome by working code through the end of this year. Core developers are skeptical of my ideas. But if they are won over by working code, then what to do with the $1.6 million daily block creation reward is moved from fantasy conjecture to pressing importance.
The block creation rewards increase with the price of bitcoin, halving in the summer of 2016 to 12.5 bitcoins per block, but staying at that schedule through 2020. Supposing bitcoin indeed reaches $1 million per bitcoin in 2017-2018, then the daily block reward in 2017 might be $1.8 billion per day.
Provided that network infrastructure is generously funded, including just say two hundred paid developers, and that sufficient dividends are paid to support proof-of-stake voting, why not subsidize transaction fees with the block creation reward? I can imagine circumstances in which bitcoin transaction fees could be negative. Visa could not compete with that. The challenge is to prevent abuse, e.g. we could figure out a way for merchants to identify themselves and subsidize transactions paid to them.