Changing the Highway Code, regrettably, is unlikely to change behaviour on its own.
For most people the Highway Code is something people "mug up on" for the driving test, and never look at again. How many people have a copy of the new Highway Code? When did people last look at the Highway Code? A possible Graham poll?
A driving licence is a strange thing, compared to other forms of licence.
It requires a significant amount of training, and the person is then in possession of a licence, which allows them to be in charge of a vehicle which could do a lot of damage if used incorrectly. There is also a significant amount of law around driving, with changes every so often.
Now in most areas like this, there is a requirement for Continuing Professional Development. A driving licence isn't so different from these, and yet it's just seen as such an everyday skill or right that there isn't the same requirement to keep up knowledge and understanding. Indeed, every so often I see clickbait to articles about how drivers are "only just realising" something that they really should know if they are on top of their game.
Perhaps there should be more of an effort to ensure drivers are up to date with changes, and are actively aware of things. It could be as simple as providing an annual update, requiring acknowledgment, or a simple theory re-test every X years. I don't think it would be popular, given that driving is often seen as a basic life skill and right (there would be more than a few "I've been doing it for 30 years and not had a crash" comments) but at the same time I'm very aware that a driver is able to wield a lot of damage if they don't know what they're doing, and really ought to be aware of what they should be doing! Perhaps it should be a condition of insurance, and paid for by the insurer, on the basis that CPD may reduce the total cost of payouts!
It happens in a small way - for example sending some on Speed Awareness courses. But that's more reactionary than 'normal' CPD!
Edit: I don't drive. Maybe this already happens and I just haven't noticed!