You yourself just said that everyone should know. Rather than creating needless paperwork for everybody they'd be better served with a public relations campaign by the government to teach people that your work email is owned by your company which means they can read and monitor it.
I'd guess it would end pretty quickly if they just stated everything you do through or on company property will be monitored as the company owns that work. While the company will often not be too concerned about the occasional personal use of their property, they aren't obligated to allow it. It gets murkier when they require access to your personal device, which I am firmly against.
Somewhere around where the slippery slope fallacy starts.
Snark aside, there's no real downside to mandating something in an employment contract that is probably already in most employment contracts anyways. This catches the stragglers.
Why don't governments produce standard draft employment contacts? They already do so for many things (e.g. buying/renting property).
Employer could ignore the standard or modify it and the employee could still amend it but it would help small companies get this sort of thing right and help with protecting both parties (which is the point of an employment contract).
In practice, lawyers will produce/modify standard contracts for extremely cheap, and as a bonus, if something goes wrong, you now have a relationship with a lawyer. If you're really cheap, in my country, you can literally go to the local book store and pick a model contract off the shelf.
I don't think this is a problem that needs solving by the Government, since these contracts are essentially a commodity and have wound up being priced as such.
You yourself just said that everyone should know. Rather than creating needless paperwork for everybody they'd be better served with a public relations campaign by the government to teach people that your work email is owned by your company which means they can read and monitor it.