I'll explain it the way I do for folks who ask me. Not everyone may agree, but based on metallurgy, this is how I see it.
A blade that has been cycled in forging should have a fine grain if all was done right. Lath martensite in steel can be considered as a long 3-D stack of blocks ( grains) - all neatly arranged and stacked in layers. If we start hammering on the steel/stack, we create dislocations - places where the joints between the blocks/grains get moved, and don't line up anymore. If we move it more, the blocks/grains get turned on their edges, making spaces called voids. These empty spaces make the volume greater, and the density less, not tighter and smaller as the edge packing and cold forging folks claim. The example I use is to take a box and fill it half full with neatly stacked blocks. Draw a line around the top of the blocks on the inside of the box. This is thermally cycled fine grain steel. Now, shake the box up good , and look at the height of the blocks. The blocks/grains are the same size as they were, but the level in the box is much higher than it was, due to dislocations and voids. No amount of pushing or pounding will make it go back to the same level. You will have to re-arrange the blocks neatly again to get it back in order and tightly packed again. This is what normalization does - rearrange the grains/blocks back to a neat "normal" order. If you hammer the blade after normalization, or after any thermal cycling event, you will create chaos, not order.
As you hammer the blade, and the dislocations and voids are created, the steel seems to get harder ( thus someone thinking it got denser). This is because the grains are disarrayed in such a way that they get jammed up and don't want to move easily. If we continue to hammer, the grains on edge may start wedging themselves between the neatly aligned grains. This creates bigger voids, and weaker bonds. Enough of this and a rift forms such that the bonds/boundaries between the grains can not hold on anymore - and a crack forms. So, a little cold forging may do little harm (and certainly no good), but a lot of cold forging may actually weaken the blade. The traditional cold forging advocates always explain that they are very gentle and use a small hammer, watching temperatures, etc., and that is how they avoid this. The actual fact is that by being judicious,they make either no change in the grain structure, or minimal bad change, and the effects are going to be wiped out in HT anyway.
As to that HT statement - Regardless of the disarrangement done by cold forging the blade in the final shaping steps, a proper HT including:
1) thermal cycling ( which is what a multiple quench really is)
2) the lowest austenitization temperature that will fully dissolve the structures
3) a sufficient quench
- will always lead to a fully martensitic and fine grain blade.
Any previous procedure that did not make catastrophic dislocations ( cracks) .... prayers, facing true north, rubbing it on your gal's boobs for luck, or waving a dead chicken over the blade at midnight, had no effect .... it was the proper metallurgical procedures used in the HT that made the blade come out well.
So, one would ask, "Why do some well known traditional smiths and/or master Japanese smiths claim to do this?" It is because of tradition being the source of their reasoning. That is not necessarily a bad thing, but can lead to mis-interpreting the results. The reasoning system may go like this - Cold hammering the bevels has been done for 1000 years, and the entire blade making procedure has been honed to the point where very good blades are made. The smiths don't know the science, or reason why it works, but they know the results are good. Someone surmises that the cold hammering works because the molecules are packed tighter together. They then claim that because the blades are hard and of good quality, ergo, the cold forging reasoning must be right. They offer the good quality blades as "proof" of their theory. If someone was to argue against that reasoning, the answer will always be, " We have been using this procedure for 1000 years and the results are always the same." What is failed to be seen, is that there may be no connection between the cold hammering and the final blade results. They could have said that the reason the procedure worked was feng shui, or the that the Forging Gods were pleased by the incense and prayers they offered before the blades were made. ( actually, that was ,and still is in some cases, the reasoning used). The smith would say, yes, the blade was good, so the Gods smiled on it. If there was a problem with a blade, they would assume that the Gods were unhappy with them - not because something went wrong in the materials or HT. As the world has become more educated and metallurgical science has become more extensively understood, things like facing true north and the dead chicken have become humorous anecdotes, but some old traditions are clung to for a long time.
So, to sum it up, there is a metallurgical reason for some things and a traditional reason for others. They sometimes agree with each other ( like thermal cycling/multiple quenching the blade for fine grain, or repeated folding to even out carbon distribution), and sometimes disagree ( like cold forging making the steel denser, or an overnight stay in the home freezer making a blade a lot harder). Both sides will claim they can prove it, and neither is likely to change their mind. As things get more modern, the new generation of smiths has more modern metallurgical training as well as traditional training, and new and better processes are developed. This will replace some old traditions, but many explanations/traditions will die only when the folks who do and teach them die.
The Japanese smith's reverence for the master who taught him will make the newer smith reticent to speak against his master's traditional explanations, so he may just keep quiet and make good knives. If asked why his knives are good, he may well, out of respect for his teacher, state what he was taught ( not what he knows). It may take many generations for these ideas to finally drift into folk lore and be replaced with metallurgically sound explanations and procedures. The dead chicken already has passed into the realm of foolish superstition, the prayers and incense are mostly gone, and some day cold forging will be ,too.