What happens if you do no cover the soil with the plastic tarp? is it really necessary?
It depends on what soil mix you use.
I compared SWC with & without plastic last year. Used a faster draining mix than you buy from Earthbox or Gardener's Supply. The SWC with plastic did better, it helped keep the top of the soil wet.
However the Earthboxes are relatively shallow, so if you use a mostly peat based mix with them. The top of the soil might keep damp without plastic. I'd try it out, if the soil top stays wet without plastic, that is probably better.
I use something similar called an Growbox and the vitamins & nutrient pack is attached on the underside of the burlap top. Without it the plants may not do as well and you would have to do the watering with feed from the top which isn't necessary with the well below.
Here is a link that might be useful: GrowBox