Well, in terms of Fighter maneuvers, the reason for more dice is simple... more dice means more things to "trade in" to do all the maneuvers a Fighter has. Cut back on the number of dice, and you cut back on both the number of maneuvers you can do *and* the amount of extra damage you can also do at the same time.
That might have been one of the problems they found with the previous packet when the expertise dice went from 1d4 to 1d6 to 2d6 to 2d8 to 3d10... each die spent for a maneuver at higher levels was an increasing amount of extra damage LOST. So balancing the two was probably harder. After all... when one maneuver was worth only 1d6 at a lower level but 1d10 at a higher one... that maneuver's cost is getting higher and higher the more powerful a PC gets (which doesn't make a whole lot of sense.)
At least by keeping all damage at d6s (and giving out more of them)... a maneuver that cost one die at level 3 has the exact same cost at level 15. Which wouldn't be the case if the size of the dice kept getting bigger over time. Yeah... a PC at level 15 might only be rolling 3 Martial dice in that scenario rather than the current 6... but if those 3 dice are d12s? Do you really want to give up 1d12 or 2d12 worth of damage just to Disarm someone? I doubt anyone would. But if its 2d6 to Disarm them, plus you still have 4d6 worth of extra damage to add... that's a bit more cost effective and will make the use of maneuvers actually occur more often.
Disarming the enemy before he can hit you with that Morgul or poisoned blade could be well worth more than 1d12 or 2d6 damage. They can tweak how often you get bumps. By removing the silly extra fighter flat bonus, and even the strength bonus to damage, they can make the system a lot simpler than it is. Strength bonus, I used to think, was a sacred cow that I didn't want to lose, but with all this damage, it's superfluous and bloaty at higher levels, and like Mearls wrote, breaks low level enemies (there is no chance to not kill a kobold at level1 if you have any kind of str mod, since you'll always do at least 2 damage).
I get your point about the same maneuvers costing you proportionately more damage at higher levels, but there is such a thing as overkill, right? With whirlwind attack, for example, instead of doing 6d12 extra damage on one guy, who'll be so dead it's not even funny, do 3d12 on three guys having sacrificed 3d12 to do that. If you're smart or lucky, you can kill all of them. That's a worthwhile tradeoff, IMO. Especially if you're wielding a sword of slaying X and you can gain multiple times the benefit of the 2d8 flame or holy damage.
The 4e motto for rangers was multi-attacks on one foe to kill'em dead faster. The DDN motto seems to be, you have the flexibility to assign damage where you want to, with some tradeoffs for splitting the damage over foes. Don't split it around if you really need the first big guy dead. But if you're fighting eight mooks and one BBEG at the same time, you really want to pick off 2-3 mooks every round, to reduce incoming damage, rather than just concentrate all your dice on the BBEG and die in the process since the number of incoming attacks stays the same. Mop up the mooks first. That makes perfect sense to me.
If maneuvers aren't worth a whole W's worth of extra damage when using a greatsword, but are worth a d6 or d10 of extra damage, then maybe if you're wielding an offhand light weapon then you are only trading off a d6 for a d6 of parry with that hand's weapon die anyway. I think that's a feature of the system, not a bug. You really should only be disarming that enemy when it's way more important to not get hit by it even once, than it is to simply finish off the foe faster. Or blocking, or parrying. If I give up d12 damage for d6 damage reduction, is that not worth it?
Maybe not when I'm at full HP, but when I have 3 hp left, you can betcha it is. I like the fact that finessable, lower damaging weapons would be used more for maneuvers than a big old sword. You can use a polearm to trip people up, since it's a ranged control weapon. So what if you trade off d12 to do it, all your friends now have advantage against him. Pummel him to pulp now, guys...ps you're welcome.
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(which doesn't make a whole lot of sense.)" there are probably other neat-o ways of addressing this issue we haven't thought of. If I get 1/2 the number of dice but higher values than now, you betcha the opportunity cost of chosing to apply it to damage over that maneuver is different. but that's what playtesting is for, am I right? It's just an idea, anyway. I like big swords doing more damage as you get higher in level.
Stealing 4e's W damage dice mechanic for basic attacks if a good one, IMO, and partly addresses this concern. e.g.. at level 6 you get 2W. It's not such a huge bump than going up by a d6. Do it on a dead level when the d6 martial damage die count doesn't go up.