You are quite simply wrong. When you negotiate severance in an at-will employment agreement, it will involve the same kinds of contractual documents you have to sign as when you are offered severance upon a sudden termination. There are usually documents relating to non-disparagement and possibly enhanced agreements about non-disclosure or non-compete that extend beyond things in the company handbook. They include specific language in which you acknowledge the employer is offering you something of material value, the severance pay, continuation of benefits, etc., in exchange for your agreement to those conditions, and generally also a waiver of ability to make claims for additional compensation.
These are generally legally enforceable, and severance is not merely a discretionary component that is up to the company. It's something you can (generally without that much trouble) actually sue for, similar to unpaid wages, if the company terminates you without paying.
I have never once heard of a company requiring a C-suite executive to get involved for creating these very routine types of severance agreements for regular employees, and I myself have negotiated severance in a standard at-will employment agreement at my three most recent jobs.