Why does 'dividends paid' show as a net asset when the value is no longer on the books?

Its not an asset - but the account does have Debit balance.
Retained Earnings is a Credit balance type account (same as Liabilities)
So the Div Paid account is actually part of RE - causing RE to be reduced.
On the first day of the next fiscal year you should JE the balance out of the DivPaid account and put it into RE. This way DivPaid will show how much was paid each year when you run the Balance Sheets