I think the main decision here is one about the time you're willing to invest, and this depends so much on your personal circumstances. You could just turf the lot, and then you'd just have to mow every week or so. But, to me that would be a shame. It doesn't solve the problems you have with screening off things you don't want to see, and it wouldn't look very good.
Now I can see why turfing would sound like a good idea, because when I look at your beds right now, I see a lot of work. In my experience with gardening, there is nothing that is quite as labour-intensive as bare soil. It needs weeding all the time - nature abhors a vacuum and will take advantage of your nicely cultivated beds to grow wild. I actually think that if you grew carefully selected, low-maintenance things in that soil, you would very likely have less work than you currently do.
So I think in your shoes, I would keep those borders and fill them with shrubs that need little more than a prune once a year and perennials. In fact, I'd even widen the one to the left so that I had room for larger things at the back and smaller things at the front. I'd be choosing some things that were medium height, so 8-12 feet tall, to provide strategic screening of fences and backs of houses (these will also give you some acoustic screening). What you choose depends a bit on the 'look' that you want, but a mix of evergreen and deciduous will give you some colour and privacy all the year round - you definitely want a couple of evergreens in front of that breeze block section, for instance. Good doers might be things like viburnum tinus (and other viburnums) choisya, fatsia, gold variegated holly, photinia, lilac, golden elaeagnus (can never spell that word), a magnolia stellata. If you have anything lower growing, you could fasten some wires to the fences/walls and put in climbers behind.
Then I'd put some flowering perennials with year-round interest in the front: you want stuff that basically looks after itself and has the right kind of colour to match your scheme. Spring and summer-flowering bulbs are also incredibly low maintenance and will give you early and mid season colour.
This still leaves you with a big lawn for the kids to play, but it also gives them something to help you with - growing together can be a lot of fun! Kids get a lot of joy out of growing things like sunflowers!