I feel pretty strongly from experience that the plastic cover is the superior solution in many SIPs and especially Earthbox. It's obviously not airtight, and up top all you need is basic gaseous exchange (created gases may exit - often created by microbes and other chemical reactions) Especially in the case of grows in which microbial activity is critical, a plastic cover offers a total light-block and excellent moisture retention, permitting the top inch of soil to be fully utilized by microbes and plant roots.
It's also necc. to have the plastic cover fit over the top of the pot, as opposed to just laying a cut-to-size plastic piece on top. This is because the cut-to-size plastic will force all of the atmo gasses - aka dry-out forces - to work only at the very edge where the dirt ends and pot begins. Focusing those forces on this small area quickly dries out that critical boundary area and as you know, with containers, whether SIP or traditional, there are a lot of important roots there picking up the moisture that condenses out on the inside face of the pot. That's not the only reason they use elasticized cover fitting like a shower cap, the other is for outdoor use to prevent heavy rains from washing soil into the reservoir, but this edge-drying effect is very pronounced if you just lay plastic on top of soil.
I've done straw, burlap, and even sand, and best results and fewest issues with a nice fitted cover stretching over the pot edges. It really releases nutrients in soil's top layer, encourages root development in the layer and increases microbe activity significantly. You can make your own with panda-plastic, facing the white side out, and instead of elasticized just cut to fit over pot and affix in place with tape or get fancy by affixing small weights. With tape make sure not to seal it up, and leave lots of untaped edges for gaseous exchange.
A lot of people use straw and other bio mulches which is fine, obviously. However I was surprised that a plastic cover became my pref, and I do want to warn again against a cover of plastic laying just on top - it will, for that edge barrier area where traditionally roots find a lot of moisture due to condensation, dry it and only it shockingly quickly.
It's not an intuitively easy effect to forsee, but is pronounced in reality. You'll see a space develop between the edge of soil and inside of pot as edge soil dries and contracts, in the very ast spot you want it drying because of high root density against inside pot surface. Even laying cardboard, fitted, on top will force that boundary space to dry out, but since air gets to more of the rest of the soil, the drying edge-effect is less pronounced.
Otter, I had trouble understanding what the issue was that you raised in the post, though I'm still trying! I wanted to share this lesson regarding laying a cut-to-shape plastic cover laying on the soil inside pot's edges. Sorry for the word walls, I will keep trying to be more succinct, and find you some pics.