I am constantly amazed that in all my decades of gardening, weeds have never ever been a problem..they are there, I know how to manage them effortlessly and I also know that a healthy bed of soil that makes healthy vigorous plants can do their part in weed control.
The sod being 2" over the weeds and weed seeds effectively smothers and inhibits anything other than the grass crop. Even major thistle or perennial roots. Mowing all by itself removes the top growth of any lucky weed to pop through the sod...remember no lower than 3"! There is no way I would have remained sane if all weeds had to be killed or pulled out of the soil before installation of sod, seed, hydroseed. I would have given up. Never spent any energy on weeds. Never. Never had problems or I wouldn't have been paid the big bucks, ha ha ha.
The thousands of acres of sod and plant beds I've designed, installed and maintained have proven to me that a healthy crop of grass, cool season in particular, maintained correctly, mowed no LOWER than 3", mowed 1X minimum per week, bagging clippings, aerating by pulling plugs of grass and soil and leaving them to disintegrate on top of the lawn. Training the grasses roots to grow deeply by proper watering practices, fertilizer knowledgeably added 3 to 4 times per grow season...will create a crop of grass so thick no weed seeds could germinate (too dark), proper fertilizer formulations for the time of year can prevent fungus and diseases during the winter.
Weed seeds are everywhere. In our soils, mulches, gravels, bird's poop, thatch, compost...weeds are just not a big deal. Even 'perennial' weeds with the ability to grow from pieces of root are simply plants. Easy to 'starve' out by snipping off any photosynthetic growth. Roots then decompose and become part of the TILTH of the soil. Easy to 'shade' weed seeds and baby weed plants, smother them with soil, use newspaper with soil or rocks...big deal. Weeds are just plants out of place. Usually a bit hardier than what we WANT to be growing. No water, no sun, no fertilizer...plants can't grow and logically, weeds can't grow.
Do not worry about getting the entire root out of the soil. If there is no top growth that root dies. No water. No sunlight. That root will die and become part of the organic matrix in the soil.
Get some professionals out to your site to give you estimates and lots of local facts concerning your soils. Free. Do not forget to call a hydroseed company. Very inexpensive and tons of pertinent information as you walk your own landscape. Free estimates, free advice.
I would be distressed if there were zero weeds..
Hope this helps. Pictures, please.
I've just now found you are in zone 9? Where exactly do you live? You must have warm season grasses, very different management techniques. Tougher actually than cool season grasses. Still weeds are not a deal breaker in the landscape or lawns. Knowing the basic maintenance practices (mowing height, watering, fertilization, aeration) weeds will not be a problem
Note: Since you have an entire year, go ahead and solarize. You are in zone 9. Clear plastic allows IR through to the soil and weeds yet blocks the heat from leaving. Clear plastic. Black plastic absorbs the IR itself not allowing the soil to heat up very much at all. Just cover your area with clear plastic. This will kill weeds but it will also kill all of the life in the soil as well. That can and must be built up after solarizing. You should be able to kill most perennial weeds, some seeds, some fungus.