If you think burning dollar bills represents useless spending, then do you also think printing new dollar bills represents useful savings?  I think you must because your dividend idea essentially prints new bitcoin bills and distributes them, not to those who did work, but to those who held stake.  You seem to think this represents increased wealth (savings).  

The truth is that burning dollars bills is not spending because nothing was consumed.  Similarly, printing more dollar bills is not savings because no consumption was deferred.  

In both cases (burning or printing dollars bills), all you are doing is changing the value of M in the Quantity Theory of Money equation:

   M V = P Q

If you burn dollar bills, then M decreases.  To maintain the economy's real output Q and velocity of money V, the price levels across the economy need to decrease.  So I would argue that you burning your dollar bills is actually a good thing for me because it will very slightly decrease the prices I pay.  It is like a small donation to all the holders of the currency.

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.