Not under current income tax treatment, it isn't. For mandated insurance, if the employer pays, then the benefits are taxable. If the employee pays, the benefits are not taxable .
More importantly, since unemployment insurance premiums can be variable, depending on an employer's layoff history, the rate isn't influenced by the employee.
Arguably most importantly, an employee-side tax would be visible to everyone on every pay stub. The bizarre inconsistencies wouldn't be hidden away.
 Such as is the case with unemployment benefits.
 Such as is the case with short-term disability.
But I'll acknowledge the part about "the rate isn't influenced by the employee".