Good points. Yes, unfortunately it seems like a lot of stuff from CC is the opposite of what it was in CT. Though to note, I wasn't saying that the sword was not important (I also wasn't saying it was; rather, I am neutral on it), just that it can be considered separately from Masa and Mune.
However, as for why the sword needed to be fixed, there could be a few reasons for this:
1) The properties of the sword sans dreams. That is, we know Dreamstone has its own unique properties, and these would be separate from whatever advantages Masa and Mune bring to the sword.
2) The sword is a symbol. People (including Frog) look at it as a means of salvation and the triumph of good over evil. The hopes and dreams of Cyrus were in the sword, not the creatures within it. The sword, then, as a symbol may have worked as a focal point of these emotions which grants it mystical powers.
3) The sword is their home. It makes sense that they'd want to have their home fixed.
Anywho, point being, the sword and Masa+Mune can be perceived of as separate things with separate needs and requirements.
Regarding the Mammon Machine providing the "activation," consider the following lines from Melchior:
MELCHIOR: The spell to energize the
sword takes immense concentration.
Bosch: The spell to make the sword activate is a
highly difficult one.
MELCHIOR: To repair the Masamune, we
must process the Dreamstone, and
activate the sword.
You work on the Dreamstone, and I'll
work on the sword.
Bosch: In order to repair the Grandleon, it is
essential not only to refine the Dreamstone but
also to make the sword activate.
You refine the Dreamstone for me.
I'll carry out the activation of the sword.
And lastly...
MELCHIOR: To energize this sword, we
must first rejuvenate the broken
sword...
Bosch: In order to activate this sword, I must breathe
life back into its substance now that it has broken
and lost its power.
First, note, Melchior has to activate the sword, not the stone. The dreamstone used for the repairs is never so treated, just refined. It seems, then, that the substance of the sword doesn't need to be activated but rather the essence of it. What is this essence? Clearly it cannot be Masa and Mune, as we see that they are indeed active and inhabiting at least part of the sword. Consider the line about breathing life. Since Masa+Mune are clearly not dead even once the sword has been broken, this indicates that it has a life apart from them (which, indeed, we see in CC).
Also consider that even after Melchior repairs the blade it isn't at full power: a sidequest has to happen for that to occur, in which Masa and Mune are the ones to make the final touch. The power of the sword is not just its substance, form, or whatever spells have been placed on it.
Regarding the Mammon machine: Melchior tells us that the activation process is a complex spell that requires intense concentration. While it is valid to say that the machine randomly had this same effect, it strains credulity.