I couldn't care less if they included CR, to be honest. I find it to be of minimal use...a very quick snapshot of a monster's overall strength as an opponent. I know many people absolutely need such a number in order for the math to work in regard to encounter design and XP budgets and so forth, so I get why it was included, but it is almost useless for my needs.
But if we imagine that it was not included.....what would happen? I suppose some DMs would simply stare at the Monster Manual incapable of knowing what monsters were suitable threats for their PCs, or how many they would need to create a suitable challenge. But for those who would not give up, what would it mean?
DMs would have to design encounters without CR, so they'd have to delve a little deeper into each monster to see if it was a suitable challenge for their players. They'd have to learn the monsters' actual capabilities and how they played at the table rather than a number that abstracted those things. The DM would have to consider if the monster with ability X was more of a challenge than the one with ability Y, which CR does not really factor.
Before long, encounter design would become intuitive and creative rather than being literally formulaic. I think in the end, games would eventually improve as DMs stopped worrying about the numbers and started paying more attention to what the numbers mean. CR and it's use in encounter design is largely, to me, a case of the tail wagging the dog.
But I know that's something that would take a bit of time. I think that including the CR was probably smart as a kind of training wheel to help people learn about how to challenge their players, but once they understand it, there is less need for it. I think the reason this edition is more loosely defined in some areas is because they expect groups to play the way that works for them. I think that the game assumes customization and house-ruling to the point where if you're still playing the game exactly the same 5 years after starting, that is more the exception than the rule.
Which I think also addresses the elephant in the room, for the most part. I've said it before in this thread in varying ways, but the game doesn't provide a solution to the problem so much as it provides examples of tools that you can use to design your own solution to the problem. Sure, you can take that idea and exaggerate it to the point of absurdity and then point out how it's flawed (your stat-less Monster Manual) but I don't think that's really much of a valid criticism.