"... looking to add some landscaping, privacy and entertainment areas ..."

Specifically, what kind of "entertainment" area are you looking to install? Pool? ...??

Please consider native trees, shrubs, plants. Birds need them. Read what the Audubon Society says. Hope you love you new home And enjoy it inside and outside.

The vitex grows well in Texas.

https://dcmga.com/north-texas-gardening/perennials/master-gardener-favorites/texas-vitex/

@Yardvaark I wish. Unfortunately, that isn't the plan for now. I just mean adding furniture, grill, tv to deck and then adding a fire pit somewhere in the yard.

Updated backyard pic with deck base finished

Screening will eat up some yard space which is a main reason tall, skinny plants like Arborvitae are so popular. But you really don't need all that much height, so consider other plants in the 12' to 15' range, too. If a central open space was circular, it would allow for greater planting bed depth at the yard corners. It makes sense to locate the fire pit centrally so as to keep it from anything combustible. Between the screening and the fire pit, there won't be a lot of room left over. This suggest the idea of making the firepit into an important central visual element, instead of just an area off to one side of the yard. It could be a perfect circle of fine gravel surrounded by a perfectly circular lawn, surrounded by a perfectly circular landscape area, the outer limits of which, of course, are the fence. The illustration is no more than a rough schematic of this concept. You could work out the details in a multitude of ways.

If you don't allow enough depth to the planting bed, you risk being forced to use only tall, skinny plants as a solution for the screening, or of making the screening into a high maintenance affair.