The rumor about not tumbling ammo has to do with the fear that excessive tumbling might wear off the retardant coating on some powders, or that it might cause some stick powders to break down into smaller, and possibly faster burning, pieces.
Is it true? Hard to say.
As some have noted, some manufacturers tumble loaded ammo with no problems. But if the cartridge had tightly packed powder, that would make a difference in the process. Ball powder might behave differently than stick powder. There are many variables.
Just because some manufacturers tumble their own ammo safely, doesn't really tell us much about other brands of ammo.
I wish I had a chronograph and especially a pressure gun, as I believe this subject is begging for some actual test data.
But, even then, if we tested a lot of ammo and found no problems, and someone asked, "But what about WWII Russian ammo, loaded at this arsenal?", we really wouldn't know for sure without testing that particular ammo.
It is a complicated question and I am not sure that there is a universal answer.