The SSA has its own statute of limitation which is different from the IRS rule of when a tax return is not filed, there is no statute of limitation.
On your IRS printout, do you see a substitute of tax return filed by the IRS for you during the time limit?
Did anyone issue you a W-2 on time, within the time limit?
Showing that you paid estimated taxes for income tax does not mean that you had reported self-employment income and paid self employment tax. They are two different types of taxes.
Because your application has been denied, in the appeal process, you may need professional, legal help. This issue does not seem to be a winnable issue. Sometimes when the two systems of IRS and SSA do matching, some years of records may be automatic updated. Wait for several years, maybe the SSA record will change. But this is not an arguable issue to make it happen.
Below two website links for your reference.
Fiona Chen, MPA, Ph.D., CPA, ABV, CFF, CITP