Most states only allow lienholders to hold electronic titles.
So it usually goes like this:
You payoff the car and have to also submit a federal odometer statement by mail or fax to the lienholder. Then, the lienholder will electronically notify the DMV that the lien has been satisfied. Finally, the DMV mails you the physical title to your address.
It’s all automated except the odometer statement, so it shouldn’t take more than two weeks turnaround.