Have you used a sod cutter? This would be the best way to get rid of the old grass and roots. I would make new plant beds with this excess material. Valuable organic matter. If the roots are below the surface by 3 -4 inches, buried by clean topsoil, mulch they will not regenerate. If they do they are so easy to pull. The sod cutter will get rid of the live grasses and the thatch from your lawn. I turn the cuttings from the cutter over when I add them to a new planting bed. Not an existing plant bed as it could cause problems with shallow rooted plants. Tilling is just not necessary to make a new lawn. Rake, grade, ROLL, rake more, make sure that your lawn drains somewhere, where you want it to drain at least 2% slope. Roll again (water filled roller works well), definitely use sod butted up closely to the neighboring sod pieces. Keep the grass of the sod from bending and getting between the sod pieces. If you make sure to mow your grass no lower than 3 inches (cool season grasses), water deeply (4-6inches deep in the soil), allow to dry before watering again (the best test is to walk on your lawn and if your footprints remain then it is time to water again, fertilize with slow release organic fertilizer (Dr. Earth was wonderful), aerate once a year minimum...you should not have to worry about lawn weeds again. Mow any lower the weeds seeds that are blowing in all the time will get a chance to germinate. Water every day, you'll be hanging a neon sign out to all weeds to come and join in...fertilization should be 3-4X per year and make dang sure the formulation matches the time of year. Especially the fall. Sharpen your blades obsessively. Never miss a week of mowing. Twice a week is great. I'd bag clippings using them on plant beds to smother weeds. But thinly. Do not use herbicide plus fertilizer. Just organic fertilizer labeled for the time of year. Tilling would do you no good whatsoever. But if you do make dang sure to ROLL and rake and ROLL some more. Do not use seed to grow grass if you've got roots and weed debris. Sod is the best option for you.