This may begin to take off in the coming years but there'll definitely be bumps in that journey.
For starters, the question of ownership will come up. Would you want a Netflix-style model where you can pay monthly and access everything, or do you still like to own the game yourself? For CDs and DVDs, most people seem to be okay with not owning the physical product, hence the rise of Netflix, Spotify and their contemporaries, but games seem to endure in that regards - anecdotally, there seem to be far more people unwilling to part with the idea of owning the physical game than was true of CD/DVDs. We saw that at the beginning of this console generation when Xbox wanted to go the frictionless, online-only mode and that was vociferously and overwhelmingly shouted down.
Then we have digital ownership, which straddles the middle ground. You rely on being able to download the game, but after that it sits on your hard drive and it's for all intents and purposes yours and you can play it when you want.
There'll be more players involved, in terms of infrastructure, especially for some third-party streaming service that might be running on Xbox Live or PSN. You're much more dependent on the online services staying online if they go offline, as they have been doing, then you're really out of luck. At least with the current model you can play your disc games and digital games (if you're on your Home console) and enjoy the singleplayer aspects. With streaming... nothing.
And what do you do if there's only one streaming service for your platform and they lose rights to a game or decide to not renew its licence? Can you never play that game ever again? From the achievement perspective, let's say you were playing a game and were at 500/1000 Gamerscore - or from the story point of view, halfway through the campaign - and the game is removed from the streaming service. If game ownership is no longer a thing and you can only stream, are you okay with never ever being able to finish the game?
For game developers, how do they get paid? Is it based on minutes streamed of their game? If 1 minute streamed = $1, this would be supported by indie devs I imagine but the AAA studios would hate it. It would mean each dev is paid based on how much their game is actually played - which sounds like a perfect ideal - but that bears its own issues.
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Streaming can be cool. It can give us access to many more games, we could try out entirely new genres, or take a risk on a game we would never have bought, and we can get that access quicker with no download times, just some initial buffering. But it has its hurdles to overcome; we need to become comfortable with offloading our games collection and not owning it, we need to vastly improve infrastructure to support it and we (the industry) need to figure out how money is extracted from that system and paid back to developers.