Get the plastic itself analyzed. It is likely Polyethylene. PE itself is harmless -- chemically it is essentially a very long chain wax. If there were any solvents in it, by now they have leached into the soil (rueful grimace). In many jurisdictions using PE mulch is allowed in organic farming. Get a ruling from your local authority.
Long exposure to sunlight will cause PE to cross link. This reduces it's strength and makes it brittle. It gradually becomes PE confetti. If you have sandy soil, this actually helps with water retention. If you have clay soil water movement can be slowed too much.
The problem with removing the topsoil to get rid of it, is you are throwing the baby out with the bathwater. You want that soil for growing your crops.
I would look at doing this:
Modify your plough by removing the plough shares, leaving the coulters in place.
Go over the field with coulters several times. You want to cut the plastic every 4 inches in two directions.
Put the plough shares back on.
Plough and disk the land.
Try harrows of various types. Chain, bar, and spring tooth. These may catch the plastic and flip it to the surface.
Try using a hay rake to see if it will gather the bits that are on the surface.
If #6 didn't work, try using a hay bine with the blades removed. (Essentially vacuuming the field.)
This will remove only the plastic that is mostly above the surface. Disk again. Repeat whatever worked before.
Rototiller may work to bring deeper stuff up.
Rock rake may lift stuff up.
response to comment on burning.
This will be largely ineffective. The grass will have created a sod above the plastic (I have had this happen in my tree farm). Grass fires generally don't get hot enough to kill the roots, let alone burn the plastic 1/2 inch to 2 inches below.
You say that it's fragile. Cheap non-stabilized plastic then.
You could make a machine to do this.
During a dry summer work the dirt up. Plough disk, harrow, rototil.
Use a haybine or peatmoss harverster and go over the surface, You want some device like a vacuum cleaner brush that lifts the top half inch and flicks it into the air. The vacuum pulls the flakes of plastic (high area, low density. high area/mass ratio) as well as a lot of fine dust (low area, but even lower mass, high area mass ratio). A correctly designed cyclone separator should get most of the plastic separate form the dust, which you return to the soil.
Weigh the plastic you retrieve. Calculate the mass of plastic originally in the field.
Use the least energy intensive way to stir the soil to get the next harvest of plastic.
