River birch tree roots cause lots of damage to foundations and driveways if planted close. Those roots extend well beyond the canopy in search for enough water. There is an oak that I have simply fallen in love with and think might be perfect for your described space. It is the Crimson Spire Oak. The only thing that would worry me about planting a row of oaks though is the root fungus killing oaks that is spreading from California and other places. Lots of beautiful and majestic oaks have died from that fungus. I do not know if the Crimson Spire can resist that root fungus.
The Natchez Crape Myrtles grow up to 30 to 40 feet tall and sometines a little higher if grown as a single trunk tree. If you plant them, space them properly for the tree's mature single trunk canopy width. Also make certain to plant ones which have already grown to at least 8-10 feet tall, and keep pruned away lower branches which might grow low enough to cause problems for people and vehicles on the driveway.
You could also plant some rose bushes beneath and between the crape myrtle trunks. Then years later if the Crape Myrtle canopy gets to dense and restricts the roses from getting enough light to bloom well, just transplant the roses to a sunnier spot, and put in shrubs which bloom well in the amount of shade that the Canopy creates. If your Crapes go dormant in the winter, it would also be nice to instead plant some appropriate featured evergreen shrubs under and between the Crape Myrtle trunks. I really like Yaupon 'Nana' holly leaf texture for growing under Tuskorora Crape Myrtles, which have reached 10 feet tall. Your Natchez trees most likely would need a shrub that grows a little taller. than that. My friend has some 'Carol Mackie' Burkwoodi Daphnes planted in a raised bed under some of her Muskogee crape myrtles and they look very good with those newly planted nine feet tall crapes.
Then again there are some Plum yews which can take more heat and will grow in zone 8. If you select a plum yew get a cultivar which will not grow above the desired height and does not spread so far that it spreads onto your driveway.