The US government is a bureaucracy, and so each department has very specific tasks or areas they govern. This is one reason it takes forever to get anything done.
But, yes according to the SSA, an SSN is not required. However, the SSA is not the agency in charge of taxation, so it would be pointless to go to them with tax questions. The IRS is in charge of federal taxes, and they are who you go to about federal tax issues, not the SSA. To pay federal income tax, you need a TIN, which in the case of a self employed person is their SSN/EIN or ITIN number.
How to Pay Self-Employment Tax:
To pay SE tax, you must have a social security number (SSN) or an individual taxpayer identification number (ITIN).
And again, you can only get an ITIN if you are foreign and are not eligible for an SSN:
The IRS will issue you an ITIN if you are a nonresident or resident alien and you do not have and are not eligible to get an SSN.
This automatically disqualifies all natural born US citizens from getting an ITIN.
You are partially correct in that the SSA does not require that citizens must have an SSN; however, it is what the IRS uses for an individual’s TIN number, which includes use in income tax and SE tax (though for self employed, you can get an EIN instead from the IRS).
So yeah the SSA doesn’t require you to have one, but you’ll have one hell of a time trying to pay taxes (among other things) without one. And yes, as a sole proprietor/single member LLC, you need an SSN to get your EIN.