The employer should provide severance. Their act of removing a person's livelihood can have devastating consequences for the individual.
The employee shouldn't have to give notice, and forcing someone to PAY a fee to NOT work is forcing them to choose between wage theft and indentured servitude.
By and large, losing one employee won't make or break a company (and if it does, it should have given them more equity and/or autonomy to have a stake and sense of power to change some of the reasons they're leaving).