I've just repaired a burn mark in a solid oak table top and it was no where as bad as yours. The mark I fixed went down about 5mm so thats how much you need to sand off minimum. The fix that I used was to let in a brass rectangular brass plate taking in the mark itself. Alternatively you can use ally perhaps brushed and use round discs instead of rectangular. You let them into the surface of the wood with a router and then glue them in and the metal acts as a decoration. Patching/ filling and colouring will have no long term durability as the colouring will degrade and end up an eyesore IMO
