Streaming, ELO and fatigue
I see several solutions:
+ Make streaming influence ELO negatively. When you are streaming, you are less focused on the game, making you worse. Something like an 0.8 skill modifier with regards to soloq skill or whatever. A 2000elo player can still comfortably win 1600elo matches, but that's it. This also prevents ELO inflation, where you try to boost a players ELO by letting him play soloq until you hit a good streak and then stopping. You will also have to invest in training lest they deteriorate.
+Make fatigue influence streaming income and ELO. Not sure how much of this is implemented, but noone wants to see a dead tired streamer. (Edit: Apparently, this is already a thing.)
+Tie streaming income to player reputation, and only increase player reputation for matches won with the player in the active roster. This approach will make the dedicated streamers less effective. A first team player should generate more streaming income and be more popular than a nobody who happens to be on your payroll. Dedicated streamers are still an option, but to maximize your profit you would need to let them play games with the team as well.