No. The SSA will not change its system to accommodate your declaration, but it really doesn’t matter to you or the rescission process.
The SSN is the property of the SSA, and remains so. No assignment of the number can be treated as legally valid, if you have not accepted that assignment knowingly, thoughtfully, intentionally and voluntarily. Therefore, the actions of the SSA do not affect your declaration of an unconscionable contract.
If the SSA’s failure or refusal to adjust their records to reflect that you have rescinded, and their failure or refusal ever causes you any damage, they may be held liable for said damage.