Its more "What does the alignment/allegiance stand for" than a what can they do.
What they stand for should be defined by the individuals, not the tools they use. Individual characters determine what they stand for -- alignment is merely a description of that in cosmological terms.
To me, its more like saying "Thor exemplifies lightning and thunder, so servants of Thor are more adept at using electricity and sonic energies." or "Good exemplifies helping others, so good clerics can channel healing spells." or "Night exemplifies the unseen and the sleep, so night mages are able tricksters" Just because an allegiance provides a standard bonus doesn't mean that this bonus is restricted to that allegiance. That said, I don't necessarily agree with the bonuses that are given for each alignment.
The problem with the above is that while Thor can represent lightning, so can Quetzalcoatl, or Zeus, or Marduk. While night can represent being an able trickster, it can also represent fear and paranoia, or the healing power of moonlight. The symbols of thunder and lighting are multivalent, and can be taken by different characters in different directions.
In the MoI system, if you want to be an extra-mobile warrior, you HAVE to be Chaotic. There's no other choice given. If you're not Chaotic, you will not be as mobile as those who are. Whereas a worshiper of Thor and a worshiper of Zeus can stand side by side and have the same power over thunder and lightning but vary extensively in details, a Lawful Incarnate can NEVER be as mobile as a Chaotic incarnate. Mobility is inherently multivalent. There's nothing more inherently Chaotic about mobility than there is about accuracy. They are both tools to the same end, victory. MoI removes this multivalence and assigns mobility strictly to Chaotic creatures.
Say clerics can get a Lightning domain for power over lightning. A cleric of Zeus may have the Law and Lightning domains, and favor the use of a javelin and spear, standing in the back to judge the combat. A cleric of Thor may have the Knowledge and Lightning domains, and favor the use of a warhammer, clashing and roaring like a thunderhead. A cleric of Quetzalcoalt may have the Lightning and Healing domains and favor the use of a dagger, staying in the back and mending the wounds of their party members.
Say there is a class that has the Tumble skill. But the only way to get it is to be Chaotic. Now, only Chaotic people can tumble, and to tumble is to be Chaotic and if you want to be an ordered acrobat who excells in finding the right place for hands and feet to go, or one who is more concerned with selfish interests and avoiding being struck for their own good than with any sort of cosmological Chaos, those become invalid character types in the game. This makes the game worse; it limits the useful and viable character types that can di a particular thing.
I'm not even entirely comfortable with what the core rules do, but that is quite mild and easy to hand-wave (a chaotic monk won't unbalance the game). In MoI, it is far to integral to the alignment to be really separated.
Does it really force those tactics? Or does it allow the character to expand his tactics? Giving a chaotic character the option to be a bit faster does not force non-chaotic characters to move slower. Giving good characters the option to be harder to hit does not force other characters to abandon armor.
But it does mean that no matter how hard the non-chaotics try, they can't catch up to him. Ditto with the Good. If Chaotic people are the best at running, the best runner in the world *has* to be Chaotic. If good people are the best at defending, the best defender in the world *has* to be Good. And if you want to be the best defender in the world but happen to be evil, well, it's impossible. And if you are neutral, well, you can't really be the best defender, the best runner, the best attacker, or the most consistant hitter.
Making Chaotic people faster means that non-chaotics are now slower. Which means that non-chaotics who focus on speed are now playing against type, like an orc wizard/sorcerer, they will never be as good at what they do as they would be if they were simply different.
While I don't really mind this on a racial level, I do mind this on an alignment level, because alignments are vague and multifaceted on purpose, and they should be.
Also, there is a difference between "any alignment can do any of that" and "any alignment can do any of that as well as its opposite alignment". By the core, every alignment can heal and harm. But good it better at healing, while evil is better at harming.
I do believe that any alignment should be able to do any of that as well as its opposite alignment. The core does introduce this problem, but it does it in a mild, relatively narrow way. Because good and evil exist as eternal tropes, it's easy to say that "good helps people and evil hurts people." But MoI exacerbates it to very difficult levels. It is MUCH harder to say that "good is a better defender than anyone else." Heck, a sufficiantly dedicated evil cleric could be as good at healing as a good cleric, and vice-versa. Give both a wand of cure wounds and watch them all do the same thing. There are questions about that, too -- a good cleric who focuses on buffing the fighter is undoubtedly better at harming than an evil cleric who focuses on dispels and debuffs. But no evil Incarnate could ever be as defensive as a good one, and no good one could ever do as much damage as an evil one. This is a problem, in my mind. It removes the polysemy from the world. In MoI, Chaotic = Mobile. If you can detect his alignment, you know his skill. And no one else of any other alignment can match him, meaning that mobility is imagined as a chaotic quality, and that a lawful Incarnate who tries to be mobile is wasting time and effort because he just won't do it well and he won't use what his own alignment gives him.
But it isn't one class meant to replace all of the major archetypes - it is a magic user (or a magic-using warrior) who delves deeper into the workings of souls. A chaotic incarnate is not a replacement rogue.
It's one class with a multitude of fighting styles. These fighting styles depend on individual belief and action, not on choice, and they preclude certain character ideas from functioning. A chaotic incarnate might not replace the party's lawful rogue, but certainly they will be using the same tactics to fight because both emphasize the same ability. And the chaotic incarnate will have things that the lawful rogue might want to enhance his own abilities, but will never be able to have because "Law isn't like that!"
Alignments have been descriptions. MoI turns them into archetypes. And I am *not* comfortable with that move.