I had plantation shutters on the sliders in a past home. They were the shoji-type where each panel was on an overhead track. Beautiful with the louvres open or closed, but when you need egress, the louvres must be closed and panels stacked to one side. The patio door frame was enclosed in a shadowbox casing to finish the look.
Here, our lanai doors are often kept open when we are home, so I opted not to use plantation shutters this time.
We presently have a type of woven wood (bamboo). Like the bamboo roman shades, but in a vertical panel that easily slides and hangs nicely. We finished it with a cornice of the same, and carried the woven wood romans through the rest of the room. They were not inexpensive, as they had to be custom sized and ordered, but they were considerably less than plantation shutters (which have a life span of forever, and are well worth the investment)
These woven shades come in different opacities, allowing for as much or as little light coming through the material as you like.
I will say, we do not have pets currently. I am not sure if a heavy weight pooch or climbing kitty would be hard on them around a patio door.
My neighbor opted for fabric panels with very large grommets in the top, hung on a sturdy, fat decorative rod. They slide back and forth easily ( no cords to pull) and compliment her room beautifully. These too, were custom sized.
I know it is more expensive, but aside from the plantations, making the covering extend as tall as possible is the way to go.
I even did that with our shower curtain. A few extra bucks, but the look is worth it.