fencegates
I had a metal gate put in on the side of my house.
I put some metal privacy screen behind it but it doesn't seem to be too private.
It just seems to add a little opacity but otherwise you can see through.
Is there any way to make this more private?
I tried painting it black but that didn't help. I thought of some random color spots but I don't think that would look good.
My last thought is to keep putting paint on it till the holes close more (or paint a nicer backyard on it).
Eight feet is a common distance between posts, due to it being a standard lumber length. You will have a stiffer, longer lasting fence at 6-7 ft. An integer number of pickets with your desired spacing is a good way to decide from a design standpoint.
I prefer to sink posts to at least frost level (42" in my area). The 1/3 advice is good if frost is not an issue.
I also prefer crushed gravel (21AA or paver base) to place wooden posts (versus concrete or pea gravel). 3-4" in the bottom of the hole (beneath post), continuing up the sides of the posts to just below the surface. Give the post a push after first 5" and tamp along all 4 sides as you fill. This technique will greatly extend post life and minimize rot.
I've used the gate brackets that you build into the gate which have hinges built in. They will stabilize a 42" gate (each side). Extra long screws on the top hinge are helpful. Stainless steel cable run diagonally (with adjustment turnbuckles) run from the top of the hinge to bottom of the gate (at the opposite corner) will make it sag proof.
Disclaimer 1 user of product, no financial connection: 
Disclaimer 2: Haven't used it but sounds interesting, YMMV (Near) Instant fence post backfill
If you don't care about how it looks, a piece of plywood or other rigid material mounted to the gate would bridge the gap. You could attach it to the vertical part of the gate using a variety of types of u-bolts/brackets.
Otherwise the easiest thing is to replace the gate with a wider one. Another alternative is to move the posts to reduce the gap on the existing gate but this requires installing posts and replacing the section of fence.
Best Answer
Your first approach of random color spots would work, but as you suggest, it might not look pleasing to your eyes. The other suggestion you have about painting a nicer yard on it is "spot on."
Anything you do to the mesh to break up the pattern currently in place will serve to hide that which shows through the mesh. Abstracts and landscapes and other images that have multiple shapes will work better than geometric patterns.
The military did some studies regarding camouflage painting and found that very little variety worked quite well, as long as it was irregular or scattered.
You could duplicate the block-work adjacent to the gate, but it has such a uniform appearance that it likely would fail to hide that which can be seen through the mesh.
The real answer is something for viewing that meets your opinion of attactive, while maintaining a broken or segmented image.