This depends on the conditions the biomass is decomposing in. A lot of carbon can be captured in the soil if handled well. No till, no spray, cover crops, and managed grazing of the cover crops can create an environment where the soil is a carbon sink and a better base for future crops.
Yes. And also no. Practices like that are good, but the majority of your carbon sequestration will come in the form of the roots of plants. (This is part of why no till is good.)
Getting surface vegetation to deposit carbon in the soil long term is trickier. You aren't wrong, but it's also not as simple as some folks believe. Just cutting the corn stalks and cobs and leaving them on a field won't put much carbon back in the soil.
