In the US most of the biomass is tilled into or left on the field to reduce erosion, and some is bundled and used as fuel in electrical generators.

No-till farming is fairly widespread in the USA though certainly not exclusive.

https://www.britannica.com/topic/till-less-agriculture

It's interesting to see how long some ideas take to gain a foothold. I remember reading about zero-till farming in the late 1980s and how it improved yields, decreased erosion and saved on fuel. Fast forward to present day and one of the farming youtubers I watched just started no-till about 2 years ago. Last night I the same youtuber posted a video about ripping out fence lines because it causes issues with their harvest and is a source of weeds in the field, and I was surprised because farmers in the uk are now understanding that those same fence lines are an important source of biodiversity for the health of their soil and crops.

New processes in general take a while to propogate.

* working in a study is different from working on every field in the country, and some things are harder to put into practice than other

* some people have the mindset of, if it ain't broke, don't fix it, which is pretty easy to justify given how precarious and time-consuming of a lifestyle farming already is

* in particular, for changes that need capital investment, the precariousness of the field means that it can take time to secure the money to spend on major improvements

In parts of the US they still very much burn the plant stubble in the fall. It creates a large amount of smog that irritates the eyes and give off an acrid smell.

Rice fields in the Sacramento valley burned just this year. As if we don't have enough toxic air from wildfires...

There are a large number of insect eggs and adults in the straw. Without burning the straw, the insects will spend the winter comfortably. Then, a large area of insect infestation occurs in the second year. It is necessary to increase the spread of pesticides, causing more environmental pollution and cost problems

How to best handle it actually depends on the region. Burying the biomass works great where the necessary microorganisms needed to break it down live in the soil. But that isn't everywhere, for instance where the ground is sodden. Like a peat bog.