The fact that your license was renewed is not legal proof of payment. The usual pattern is that a court will notify the DMV/DOL in your state that you failed to appear or respond, and then under some law (e.g. Washington's RCW 46.20.289) your license may be suspended. In Washington, the law says "shall suspend"; in Florida there are multiple rules of the form "shall suspend" and "may suspend". Assuming that your state has mandatory suspension, it would be predicated on the court notifying the license department of the unpaid fines (the license department doesn't directly know that you have or have not paid a fine). While you would reasonably hope that the court would provide timely notice to the DOL of a failure to respond, it is unlikely that there is a law that absolutely compels the court to give notice within a certain time period.
Even if there is an error on the part of the court (failing to give timely notice to licensing), that does not override the requirements of the law, unless there is a specific provision (e.g. "if the court fails to notify DOL of a failure to respond within 3 years, a license may not be suspended"). An error by the court is not proof of payment. Moreover, an error w.r.t. mandatory suspension could be due to the licensing department failing to revoke upon notice.
You're trying to argue a particular fact, that you did pay the fine. You evidence is based on the premise that the court would without fail have notified licensing of the non-response, and that licensing would without fail have suspended your license upon notice. Neither of these premises enjoys much factual support (government agencies know that government agencies can fail to do what they are supposed to). Only proof of payment is proof of payment (and I assume there is no proof, in the form of a credit card payment or cancelled check).
This does not mean that you can't plead to the judge that you did pay, but it's unlikely that the court would take the fact of a renewed license to prove that the fine was paid.