As mentioned, i think Blizzard needs the streamers for publicity, news distribution and hyping the changes in each patch. Its become an important vehicle for their customers. We the public will go and see what quin says rather than search for some half-decent thread on blizzards forum.
If i try to look at this from Blizzards point of view i don't see any problematic effects aside from people thinking that botting puts you in the top rankings. The thought of that prevents others from wanting to compete but its not actually true. 99.99% botters are not concerned with grift rankings. For those that are, its only because its easy and it's not the reason they are ranking high. RNG of rift layout and monsters is more like 60% of the difference in rank, the 35% is skill at playing, researching and 'utilizing' mechanics, 5% is gear. For the amount of time played by top streamers they will already have pretty close to the best gear without any botting.
With no auction house on a single player game where you cannot trade items, what difference does it make? And there are benefits to blizzard from D3 botting.
1) Being engaged in the game exposes people to cross-game advertising and promotions and keeps their mind within the blizzard ecosystem - how many paid for Blizzcon, or bought heroes of the storm who otherwise would not have?
2) Many of us have purchased 2-20 legitimate copies of the game.
3) Accessibility, we have had posts from people with legitimate physical and physiological disabilities who could otherwise not be able to play and enjoy the game.
But lets say botting was completely stopped, this would not solve the problem i am describing and not prevent streamers from farming the gear. They earn enough money from streaming to hire someone at minimum wage to play on their accounts to farm.