Will a 3rd-year extension for Drill Sergeant benefit me?

If you love what you are doing, do another year. It won't hurt your chances at SFC. You have to take charge of your career. No one else will.

Will your branch consider an extension once you have assignment instructions? Usually AI overrule any personnel requests.
Assuming your branch will honor your request, is there anything you can accomplish in your third year you didn't accomplish in the first two years? Your branch manager doesn't know how many squad leader positions are filled, the branch only assigns you to the brigade. It's up to the brigade to manage it's strengths.
I would go back to the line, but if you really love your job, stay another year. Sometimes path to promotion is doing what you love.

Hey brother, first of all as being an Old, Crusty, Retired E-7 from this Army that I loved and served, you have to make decisions on your gut feelings and trust that all will be well. As a Drill Sergeant, yes you look great as I respected my Drill Sergeants and still remember them (1987) Go with your gut brother and I am sure that Mama want's you to get off the trail so you can be with her, but i appears to me that she is an Army wife and she will be there for you no matter what!
The life of a DS is stressful on the Soldier, Mama and the Private's, SFC is in your vision, do what it takes brother as you have sincerely earned it!! I have a CIB, but I always wanted the DS Badge that I let people steal my dreams, please do not allow that!! Trev