"In the tropics, biochar increased yield through liming and fertilization"

So it would be better to just use wood ash, which is an abundant waste product.

No, wood ash is a completely different product that is low in carbon.

In normal open air burning much of the carbon in the material is released into the air. (Not good for the environment.) Instead of ash, you want to produce charcoal - a black super-high carbon material. This is done through a process called pyrolysis. Once you get combustion you starve the process of oxygen, so you get a slow burn for many hours or days that results in charcoal.

Large producers have special ovens they use for this, but you can do it yourself. This technique became well known because native farmers in Brazil did it for thousands of years. You can start burning materials in a trench, then once it is hot slowly cover the material with a couple of inches of soil which will cut off the supply of oxygen.