To the OP, what is the end game after all the Bitcoins are distributed? You state that block rewards are meant to be a form of distribution, and they are that, but not limited to just that. They are also a reward for miners to cover their expenses. And right now they have A LOT of expenses. It works out to at least $3 a transaction.
Here is how to figure it out. The most efficient mining operation in China is producing bitcoins at $80 a piece. Lots others are of course have to spend far more than that to get a Bitcoin but lets assume the best case scenario that miners are willing to mine for $80 a coin.
Each day there are how many Bitcoins produced? lets see...... 6 blocks per hour times 24 hours = 144 blocks times 25 coins per block = 3600 per day
Now the most efficient and best miners need at least $80 per Bitcoin to keep the doors open. So 3600 x $80 is $288,000 per day. Now on a very good day (again best case scenario) we will have 100,000 transactions per day. That means the price of each transaction being paid to miners is at least at a bare minimum $2.88. In reality that number is actually much higher than that. That is where the $10 figure comes from.
Now someday there will be no block reward at all. Who then is going to pay the miners their standard $2.88 - $10 fee? One would assume at that point the only place it could come from is the users paying fees.
So therefore at some point in the halving of blocks game we reach a turning point. That is the point where block subsidies no longer cover the price for all those hashes. Again this is inevitable because sooner later the block reward is 0. And even by the 10th block halving it is under 1 Bitcoin per block. So if the miners don't get their cash then they shut down their hashers and the network dies. We all know that miners are selfish and are there for the profit. Take away their reward and they leave.
Where is your explanation and theory about BTC to Fiat price increasing over time?
The Halvings will have an effect of making BTC more rare, thus price should increase to compensate.
When price increases, this "problem" will not exist. When there are no more "rewards" (which is an incorrect term), tx fees will be mandated.
And if the price doesn't increase atleast 100% in the next four years, then Bitcoin has big problems.