1. Forging is better than stock removal.
Forging steel can allow a great way to move lots of metal quickly into whatever shape you need for a blade, so at the manufacturing level this can be true. However, in the context that most blade smiths stake this claim, in that the forging process improves the overall quality of the blade, this is not true. Many knife makers claim that forging will align the grains with the edge they are making and will improve strength or compress the steel, and while forging can align the grains of steel to an extent, this doesn't make a difference in the finished knife because all steel grains are completely redissolved during the heat treatment process anyways. If a knife is made in which the grains are still aligned in the finished product, then there was an error in heat treat that will create a much more detrimental effect on the health of the knife than the aligned grains would be able to compensate for.
Note that there is one possible exception to this first myth, and that is in traditional wootz steel. As there is so little verifiable information on this unique kind of steel it can be hard to verify, however as wootz steel is a dendritic formation of carbides in a steel matrix, cold forging with no heat treat is one theory of getting the greatest potential from this kind of steel. As soon as I am able to get my steel melting furnace working I will be able to attempt my own experiments with this kind of steel and will post my findings here.
2. Cold forging steel refines the grain size
This is a myth that is somewhat based in reality. When a steel is forged at too low of a temperature, the physical deformation created by hammering away at the steel creates dislocations in the steel matrix. These dislocations, if you remember from a couple pages ago, are part of the reason for bainites greater toughness, and so it seems like there would be some fact to this myth as grain refinement creates higher toughness as well. However the immense amounts of dislocations created in forging can create such stress in steel that internal fractures can be created that could have detrimental effects on the finished blade. Now as I said, there is some truth to this myth, and that is in those same dislocations. Remembering back to the pages on heat treat, when the steel is quenched or cooled below critical, the grains each nucleate from a different point in the steel each time, sometimes from carbides, sometimes from the alloying ingredients, and in the case of cold forging, when done properly, from those dislocations themselves. So in a way, while cold forging itself doesn't refine the grain, it can be used to set up the steel for a more refined grain later in the process, however every study done has found this effect to be almost completely negligible in the finished product.