All of the above. Some of it gets bucketed onto the compost heaps, but most gets drained onto the paddock. We have acidic soils here so the high pH is a bonus for watering, but I dilute it about 25% so that roots don't get a shock.
I quench it twice: Once to put out the fire, and a second time when I put it in a ceramic tub to cool it off. I don't want it to smolder for weeks. The first, fire quenching water just drains off. With the second, ceramic tub water, I pour it off onto alkaline-tolerating plants before crushing the biochar.
Yeah, it's tricky. Depends on the general pH trend of your soil. Some plants will put up with the high alkaline to get the phosphorous though. Raspberries, in my experience.
I, unfortunately, don't have a way to collect the quench water from my biochar production. This past time I had quite a bit of ash buildup that I did my best to wash away from the char. I imagine the quench water was mixed with plenty of ash and would be on the basic side of the PH scale.
I wonder if it would be appropriate to utilize the water in places where you wanted the benefits of 'remineralizing' as ash is leftover mineral material if I understand correctly?