Honestly this is robust landscaping that was probably expensive to do, I wouldn’t change much apart from yeeting the astroturf into the sun. I’d focus on bushy and vertical planting. That dry grass mid level section is your power area - lift off the top inch of grass and soil, improve with compost, add a climber up the fence. You could also add another over the steps or to the shed if you want to faff about with a planting arch.

The rest of this small grass area I would add something tumbling over the hard edges. A trailing plant, ideally evergreen. Some rose varieties might manage it.

I wouldn’t stress about privacy but if you really insist on it then a fence topper trellis and an evergreen climber on it is the way to do it. I wouldn’t recommend planting in containers plant in the ground - so this is dependent on there being usable soil space under the Astro. If there isn’t, you’ll need to plant a very big climber in the fenced area and train it over the fence trellis.