Answer: Probably the most common reasons for lack of root development are either inadequate soil preparation and/or lack of water. The root veggies you listed tend to do best with ample moisture and a dry soil will dramatically limit their root formation. (Radishes also really prefer a cool soil to grow their best.)
Sometimes you will find that plants will not root very deeply if the season is a dry one and you are watering because of lack of rain, and your watering is not soaking down deep enough. The roots will not grow into dry soil, so they stay near the surface where the water is being applied. You might want to check on that by watering, waiting a bit, then digging down to see how far the water actually reached. Sometimes it is a surprise!
Also, roots will grow best in a soil with a loose structure, meaning lots of air pockets and lots of organic matter. This kind of soil can hold ample air and ample moisture at the same time and this is what roots love. The roots can also grow through it easily since it is loose. You can develop this by digging deep to loosen the soil, then working in lots of organic matter such as compost, and if your soil is clay-based, possibly add some sand as well. This type of generous soil preparation combined with correct watering and using a layer of organic mulch several inches thick should encourage good roots.
Next, try growing varieties known to do well in your local area. Local nursery personnel or your county Extension or experienced gardeners in your neighborhood should be able to make suggestions about specific varieties.
Finally, be sure to augment fertility as indicated by the soil tests since veggies tend to like a rich soil. You might also want to check the pH. I hope this helps you trouble shoot.
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