Freeze just the things involved in the hit. I've always thought the idea behind hitstop is to emphasize resistance. Imagine swinging a knife. Now imagine swinging a knife through a small branch. Even if the knife is sharp enough to cut through the branch with just a swing, because the branch is solid the knife will slow down as it moves through it thanks to some resistance. tl;dr: It shouldn't affect anything else on screen but that's the easier thing to do. For a single player game, it doesn't even make too much difference.
To avoid things getting put repeatedly in hitstop (a player in between two enemies that hurt with touch idling or whatever) some games just have a maximum number of frames an object can be in hitstop before they ignore things trying to add more.
Hitstop is usually something very small like <6 frames, so you don't need to worry about the attack happening on a specific key frame. Any point during the knife swing hitbox will apply it to that frame of animation, and it should look fine. Here's a thorough article about Smash Bros. hitstop:
http://sourcegaming.info/2015/11/11/tho ... vol-490-1/