What you're describing is a car that eschews defensive driving in favor of offensive driving. Speeding up to keep people from cutting in front of you only works until the other driver doesn't see (or doesn't care) and now you're in a crash. Or traffic suddenly slows as you're speeding up to close the gap, and now you can't respond quickly enough and now you're in a crash. Letting someone cut you off and slowing to return to a safe following distance is by far the safer option.

There's a reason it's called a safe following distance. Anything closer and you cannot stop in time. Even temporarily. Especially when autonomous cars in front of you may be suddenly braking as well.