Should I make payment on a debt whose SOL is not there?
The negative item will automatically drop from your credit report after 7 years. There is no need to send any written application after the period is over. However, if the listing appears on your credit report even after that period, then you can dispute with the credit bureau.
Can you pay the full amount on the credit card? If you can't, then it's better to not make any payments. Once you acknowledge the debt and start making payments, the SOL clock will restart. Paying off your delinquent won't help to increase your credit score immediately. You have to add positive items on your credit report. Moreover, as the debt is quiet old, so it's not affecting your credit score too much now.
Yes, I agree with Nick. But, I feel you can show your eagerness to repay your loan to your future creditors if you can repay the amount in full. In addition, it will create a goodwill with the bank or financial institution from whom you took out the loan.