I think there’s minimal factual information regarding biochar function increasing fertility.
We do know a few things though. Fresh char contains strong bases, so it will naturally cause PH and nutrient flow upset in sensitive systems. This probably explains the issue with the potting soil example above. In most agricultural land the natural buffering will easily negate the change in PH.
There seems to be a lot of mystique around composted biochar, and unsupported claims aimed at marketing and profits. Of course marketers want to substitute as much cheaper or inert material as possible in their products. Mixing with compost probably also helps eliminate problems at the retail level with people using it in plant pots, washing and buffering issues, apart from the fact char is pretty inert, and not really beneficial in pure form, like perlite or vermiculite.
In many soils char will help retain moisture, and certainly is porous, forming habitat for soil organisms, this could indirectly help with nutrient availability. The presence of carbon may influence nitrogen cycling in the soil.
For sure it’s a carbon sequestration strategy, probably of modest fertility enhancement, the largest effect may be water retention. (Which will also assist nutrient availability in arid conditions)
Some studies have found fertility reductions in soil around 155 T/Ha.