The pivoting swingy-screen sounds like a great idea, but it would take some assembly, and also possibly some strength. (Perhaps better for compost and archeologists than the relatively large volumes of clay and rocks I try to screen.)
I just prop big steel wire shelving up on cinder blocks or bricks and a stump, with adjustments to change the angle.
The closest I come to that slick pivoting-rocking action is my very final clay-screening step - when I'm working with a fine 1/4" screen almost horizontal, and I'm trying to get every grain of fine stuff possible to pass through it. I tie 1/4" hardware cloth over some steel wire shelving, so the shelving wires take the weight and support the hardware cloth.
I throw 2-3 shovelsfull of partly-screened, partly-rubbed clay onto the 1/4" screen. I'll break it up as much as I can with the back of a steel rake, and rub as much of the clay through it as I can. This is tedious. Sometimes I'll do this while sitting, since my legs are weaker than my upper body.
The pivoting-rocking comes after I've broken the clay up as much as it's going to be broken by rubbing. I grab the low edge of the screen and bounce it up and down several times, where the springiness helps bounce the clay clumps around and encourage fine stuff to fall through. I think this is faster than pushing it around with the rake, because it unclogs the screen and uses the entire surface area instead of just the spots where the rake os rubbing.
Then I put the low end back down, and lift and bounce the whole screen from the high end to fling gravel and big hard clay balls AWAY from the screening area.
It takes around 10-15 shovelfulls to make the little hill under the screen tall enough to touch the screen. I use a spade somewhat delicately to scoop up the little round clay balls that went through the screen and tend to run right down the sides of the hill and collect at the base.
They seem to me to be "purer clay" or "harder clay" than the stuff that broke up easily, so I scoop that away from the hill and demote it to an "extra-bad-clay" heap to work on some other year.
THEN I push the hill around so it isn't as tall, so I can screen more clay.
Eventually the whole area under the screen is full, and I either shovel it to another part of the work area, or move the screen and start a new hill on another side of the stump.
If I'm only planning to screen a small amount of clay that day, AND have enough compost, pine bark and coarse sand, I sometimes amend as I screen.
4-6 shovelsfull of clay.
2-3 of compost.
1-2 of crushed rock.
1-2 of pine bark.
Repeat.
But usually I do it in stages: screen as much clay as I have room for. Then acquire compost, bark and sometimes sand or crushed rock. THEN mix.
(This isn't enough amendment for GOOD soil, but it is enough for SOMETHING to grow and mellow the mix through adding roots and soil life. I keep adding compost each year.)
