Answer: Go ahead and plant over the old roots. There are no diseases that will transfer from the redtip photinia shrubs to the vegetable or flower garden plants. You may wish to remove some of the shrub "stumps" and largest roots to make it easier to rototill or spade the soil, but it isn't necessary.
If you add a new soil mix that is different from the existing soil, mix the soils at the interface between the two a little bit for best results. A spade can be used to lightly break up this "transition zone" after a few inches of the new soil has been added. Thorough spading or rototilling is not necessary.
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