I'd take a bit different approach and redo the entire backyard, getting rid of the raised planters which look very arbitrary in their design and placement and excuse me for saying so, but very amateurish as well. Heavy clay soils are very common in the PNW and relatively easy to fix or at least work with. Get rid of the sand/gravel, get rid of the raised beds and import some good compost to work into the soil. Make a few easy mounds along the perimeter - longish, irregularly shaped berms, maybe 12" in height. Slope gently back to ground level at the fenceline. This will give you some additional height for any screening plantings (you can easily move and relocate the few existing plants), plus it will work to improve drainage or at least provide a fast draining planting base. I'd disagree about the Viburnum tinus, also. That back garden is pretty restricted and with the fence, should provide more than adequate protection from winds for Laurustinus, provided you are truly a zone 8 (specific location?). I'd hesitate to add any trees in such a tight situation, but some very columnar evergreens could work (i.e. Chamaecyparis nootkatensis 'Green Arrow', 'Van den Akker' or 'Strictly Weeping'). Bamboo could work also but IMO tends to provide much more of a mass than such a small space warrants. Add back the paver pathway if you like and fill in with lower plantings of your choice and groundcovers. btw, your willow "tree" is just a shrub form of Japanese willow grafted onto a standard......it will never get much taller than it is now except for some expansion of the canopy.

Landscaping a backyard for privacy means placing tall, screening shrubs near the offending property lines ... not close up to the house and patio. (There, you'd want to use low plants or overhead trees, or some combination thereof, between you and the view.) For the screening shrubs, you'd need to account for the varying light conditions, which means using a type of plant that can accept sun or shade, or using a variety of plants, each which accepts the conditions in which they are placed. Since you have a length of property to screen, it makes sense that you would place all the shrubs in a bed and keep them separate from the lawn. If you wished to incorporate any small tree forms in the mix, you might want to place groundcover below them instead of only mulch. (It's an opportunity for adding another element of interest. It could include groundcovers that bloom.) Since you want to screen the CL fence, too, it will be an L-shaped bed. If screening both CL fences, it would be U-shaped, following the fence and lot lines. The bed depth, front to back, should be sufficient to accommodate the shrub dimensions, which is going to be quite large ... so probably 8' or 10' depth. Understand that whatever ideas you are given and wish to contemplate, and possibly incorporate, need to be put in the form of a plan before you do any actual work. So prepare a base plan on which you can work out the ideas. You could begin that by tracing over a plot plan of your yard. If you don't have that, measure and draw out on graph paper.

Thanks everyone for your answers and picture examples. They were very helpful. My patio is 15 x 20 and I have found it hard to find a premade pergola kit in that size. I’m actually wanting there to be no “ceiling” to the outdoor room so we can look up at the stars (and also so we can have a fire pit). We will use umbrellas over the dining and living areas when needed. The posts would just be used as supports for curtains, string lights and to add vertical interest by growing flowering vines. I think I’ve found my solution: 6x6 cedar posts secured to the patio with the anchors in the picture below. It looks pretty straightforward so hopefully we won’t mess it up. Then I can add curtains rods and curtains.

You have some tough decisions to make about your deck or patio first. Start with your space needs (table size, firepit, grill, etc) and how you want to circulate from the house to it. Then design the softscape holistically with the hardscape (e.g. not only so they work in plan layout, but in addition to plantings at the easement line, maybe you want a partial screen or pergola built on a wood deck, or a planter box with some smaller sitting-height plantings with a patio). It looks like there is plenty of height to your first floor. A deck performs ok if it is at 16"-18" or so above grade, which should allow around 8" of air circulation underneath (which when lacking is the downfall of wood decks at grade). It would be good to take advantage of not needing guardrails (verify your AHJ max deck height) because with the basement stair guardrails and fall protection over the stair that would be a lot of visual clutter. The decisions involve where you access it, from the Family Rm only, wrapped around the corner for access only/also from the Mudroom in the same elevation, where it steps down to grade next to the sidewalk/garage, etc. A paver patio looks to have some tricky decisions as well. You never want to raise a patio more than a couple inches above grade anyway (the higher you go, the more expensive and probability of settling), and you have the fireplace bumpout which would hamper that anyway. You would have steps down from the Family Room, so if you don't want railings beyond 2 risers you have to have large landings in this small space. It also does not look like it can be at sidewalk elevation, so you will have a "3-tiered" space - porch, patio, and sidewalk - so plan for more steps and/or retaining walls. Also, the tree specimen looks like it would be problematic long-term that close to a patio. For a privacy plant screen, use your site plan to plot dimensions of mature-sized plantings. It seems like you will want good height if you have a deck and from cars approaching from the higher street. But from the pics (which might be deceiving), it doesn't look like you can fit much between the swale and driveway or swale and basement stair (and any roots would need be kept far enough away from the swale to not be sitting in wetness).