Mould starts where the oxygen is — then it grows down into the cut
A fungus needs three things: something to eat, water, and oxygen. On a cutting board all three arrive at the surface, which is why the mould always begins there and never in the middle of the wood. What makes it hard to remove is what happens next. The thread of the fungus grows downwards along the cut, following the vessels of the wood or the gap between two glued strips, and it can get one to two millimetres below the surface within a few days.
And the black colour is not the mould. It is melanin, the same pigment that colours skin, plus the dark spores the colony releases. That distinction matters more than it sounds: pigment is a chemical stain in the material, not something sitting on top of it. Bleach will lighten it. Bleach does not reach where the fungus lives, because hypochlorite is consumed by the organic matter of the board within about half a millimetre. So the board goes from black to grey, you feel that you have cleaned it, and the colony carries on underneath. That is worse than doing nothing, because it removes the evidence.
Three places, three different answers
On the surface, a fresh patch of white or grey-green fuzz that a fingernail will lift, is genuinely removable. Coarse salt and half a lemon give you an abrasive and a weak acid in one go; work along the cuts, rinse, dry immediately. Sanding half a millimetre along the grain with 120 then 220 grit takes the stained layer off properly. Then oil it with food-grade mineral oil or a mineral-oil-and-beeswax mix — not olive oil, which oxidises, goes tacky and feeds the next colony.
Inside the blade grooves, where the specks only appear in lines and will not move, you can still sand — but the board is 25 mm thick and the safe limit is about half a millimetre, so there is exactly one attempt. If the specks are still there afterwards, the honest move is to demote the board: vegetables and cooked food only, never raw meat, and never fish. A board that still has a living colony in it is a permanent reservoir, and the thing you should not do is keep using it for the food most likely to carry something.
Along the glued seam of a bamboo or butcher-block board is the one that ends it. Those boards are strips of material glued into a slab, and the glue line is the weakest, most water-hungry part of the whole object. Water gets in, the adhesive hydrolyses, and the mould follows the line into the middle of the board. Cut across the seam with a knife and look: if the middle layer is pale, soft, and you can pick bits out with a fingernail, the glue has gone. Sanding cannot reach inside a glue line — you can only sand the two faces either side of it. That board is finished.