How about GE cover crops?

In a past post, I argued that killing a cover crop with an herbicide was better for building soil than killing it with tillage. Here is another option. Why not develop genetically engineered cover crops that die easily when sprayed with an innocuous substance?
Such cover crops would be a benefit to farmers and the soil. Farmers could use these cover crops to build their soils without worrying that they would undo their progress by having to use tillage to kill the crop. An easy-kill cover crop would make a near-organic no-till system much more practical. Depending on how it was done, herbicide use could be reduced or eliminated.  It might even be possible to place such a GE trait into multiple cover crop species, to be used by farmers in varying cover cropping situations and climates.
A GE cover crop would avoid many of the objections to GE food crops.  Since the engineered organism would not be harvested, the risk of it mixing with non-GE crops is greatly reduced. As a cover crop, it would never get to the consumer – no Frankenfoods here.
Unlike current GE crops, the trait given to these cover crops would not be beneficial to the plant, evolutionarily speaking. The trait would be more like an undesirable mutation, but one that we designed to be of advantage to us. There would be no risk of developing herbicide resistant weeds. Even if the trait did escape to other species, it would be of no advantage and so would not spread rapidly.
As to how this could be engineered, I can only speculate. The ideal would be to modify the plants to be susceptible to a low dose of a cheap substance like vinegar, which organic farmers already use for weed control. Another option would be to make the plants highly susceptible to very low doses of a current herbicide, preferably one that is relatively non-toxic, non-mobile, and readily degraded in the soil. Glyphosate would be a good candidate if not for all the weeds that have become resistant to it.
I found few references to efforts in this area, although with strategies different from mine. In one, researchers engineered plants to die at high temperatures. Although they were successful, they thought that using photoperiod as the trigger mechanism would make the timing of the crop death more reliable than temperature. Another strategy, this one described in a patent held by the Idaho Research Foundation, uses hybrid lethality (also called hybrid necrosis), here in wheat, to make the crop die early, before completing its life cycle.
There would still be challenges, like what to do with the weeds growing in the cover crop, which would not die when sprayed, or when the cover crop died.  The cost of the seed could also be a problem. Farmers typically want low-cost seed for cover crops, but whoever developed this trait would want to recoup costs by increasing the seed price. Perhaps this is where public research could step in and provide a service.
I can imagine other uses for such crops. Farmers could plant them as designer weeds to cover the soil, mimicking the role of weeds and competing with them, but then spray them out before they compete with the cash crop. The trait could be used in species that have allelopathic properties to improve their competitiveness while they are growing and after they are killed. Other possibilities are genetically engineered mustard cover crops designed to produce high levels of the desired glucosinolates in the roots or high dhurrin-producing sudangrass for control of nematodes.
There will still be those who object to GE plants in any form. However, for those whose concerns are specifically with using GE crops for food, this may be acceptable. The linking of genetic engineering with soil building cover crops could be a different model for future genetic development.