Most titles that try and develop some easy publicity will just throw money at the game regardless of returns, there was a $100,000 tournament for Evolve a game that was plagued by negative reception for its stupid pre order system, it still shelled out cash just for a tournament.
I feel this is what Blizzard will also do but a lot more aggressivly. There will be people that they cannot convert over which will probably be the traders and those that have spent a stupid amount of money on the game.
The point you made on what Valve will do I 100% agree with, Valve give the impression of a company that will be pretty chilled and relaxed and then make a push for change when it's really needed, and as Valve aren't really a company that has any competitior pressure I to feel like they'd most likely wait for Overwatch to happen and then react to it than to make a move first.
There is a potential that MM in TF2 can change that, if it's done well then that can spark interest from other parties and general public players as well. One of the reasons why Overwatch could push against TF2 is just like other games. It has a ready made competitive match making mode. It's one of the reasons why I don't play TF2 as often as I used to because I can just hop in to a ranked Dota2 or CSGO game and poop on nerds.
Also as far as I know Blizzard have a more hands on approach with their titles now, unless somethings changed they now hold World Championship tournaments themselves instead of 3rd parties having control, so that's why I think that with Overwatch you don't really need 3rd party interests to make sure there's money flowing through.