Only in theory. The reality makes far less sense.
Here in California, it's not the employee who pays the tax, but the employer, and it's only proportional to income up to $7000 per year (yes, year, not month) per employee. The payout is only proportional to earnings up to around $50k annually.
What's the difference?
To the employer it's just added to the salary, they don't think about it any differently.
You could give it to the employee first, and then take it from them if you like. It's identical.
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".