In my experience, having stuck with CR religiously, you find yourself with tense, anyone-could-die sessions - against a pack of wolves from a random encounter. And then you have easy, bring it home encounters - against the arc's big bad. It takes control of the fight's narrative stakes out of the DM's hands, and makes it more-or-less random.

I want to stress that it was fun to play this way, but eventually myself and my players longed for more dramatic final encounters, and so I had to homebrew creatures.