Grass of any sort is often difficult to kill. If it is sod, you may be able to remove it for composting, Brandon. I've made some of my nicest compost with lawn grass sod.
Simply because it is covered with soil will not mean that the grass is dead or dying. However, you can bury Kentucky Bluegrass under about 8" of soil and it will die. Don't count on that with something like quackgrass. That stuff will require multiple tilling or, better yet, removal.
Winter rye can be almost as difficult to kill in the spring as any lawn or weed grass. The key for me was to allow it to grow enough that it could be pulled by hand. Once again, this grass can make good compost. But, I am very inclined to dig out beds in the garden and bury compostables - like that winter rye.
Most of us like to have soil that is easily penetrated by garden plant roots to a good depth. If your topsoil is, at least, 8" deep, it really helps if that top 8" is nicely prepared for the crops to grow in. Burying organic matter may work better in my gardens than elsewhere. The soil there is really beyond "sandy" and all the way to "gravelly." Yes, water drains well and there isn't much of a chance of just anaerobic composting going on under that soil. If I've put something like lawn grass or winter rye in the bottom of what amounts to a trench in the spring, it is almost impossible to find anything identifiable the following year. Permanent paths give me some place to pile the soil and moving it on and off helps keep the paths free of weeds. I can till the paths now and then but tramping up and down on them with my big feet probably does more to keep them weed-free than anything else.
We are each other's harvest; we are each other's business; we are each other's magnitude and bond. ~ Gwendolyn Brooks