I had a very successful composted-charcoal tomato plant experiment last year.
I had just started using charcoal in my compost piles as a carbon source the previous year, and this was the first use.
First, I make my charcoal by burning soft and hard wood tree clipping on my property and dousing the fire with water after the coal bed builds up. I'm hoping to get a little "steam blast" action to create more pores and lighten the charcoal.
So, burying the lede here, I planted a tomato plant in a 1.5ft deep hole completely filled with my compost-charcoal mix. It proceeded to thrive through a drought we had WITHOUT WATERING. It completely blew my mind. I woke up every day expecting it to be dead. But instead it fruited. Harvest wasn't huge and tomatoes were small, but the lawn around it was dead, dead, dead.
End of season I pulled up the plant and the roots had completely interwoven and penetrated many charcoal nodules which came up with the plant. I was surprised just how firmly the plant was hanging onto that charcoal. So, I think there's zero argument that the charcoal helped make it a better mix. The only question is how much of the boost was from water-holding, nutrient storage, and symbiotic bacteria living in the charcoal. I'm also speculating the charcoal was able to grab and hold water from the brief rain we did have during the drought.
I found a article last year where some university had tested charcoal amendment along side compost and the compost-with-charcoal technique. They found a 10% yield gain when charcoal is part of the composting process vs just adding it later.
Also, I'm wondering if charcoal is speeding up composting itself. I'm in Massachusetts, so our our summers get pretty warm. I process/chop all new greens with a liter or two of old compost to inoculate it and increase the surface area exposed to bacterial action. I then cover the batch with some old material.
Starting in June, a fresh load will be fully composted in 5 to 7 days. I don't know if that's normal, but it sure seems fast. I'm wondering if it has to do with charcoal helping with aeration and/or easy carbon source from charcoal dust.