I am challenging the general concept of alignments and the way they are used in most OGL settings and games.
Next to getting rid of Vancian magic, this is probably the most common challenge made to the traditional AD&D framework. Honestly, I find it a bit trite and usually misguided.
But what do they offer to an experienced player?
A great deal more than they offer the neophyte.
Firstly, there are 9 alignments. In reality, there are more than 9.000 ways of thinking and types of personalities.
So, your first misunderstanding is to equate alignment with personality. Alignment is not the same as personality and makes no attempt to record a characters personality. We could make some broad generalizations about personality by examining the combination of alignment and the characters social attributes (Intelligence, Wisdom, Charisma), but even then there would be significant room for variation and a great many unrecorded factors. Whether the person is lascivious or chaste, outgoing or introverted, fastidious or slovenly, for example, is not something that can be determined easily by alignment. Certain sterotypes suggest themselves of course, but alignment is broad enough to incorporate departures from the sterotypes if we employ or imagination.
Let's think of a woman who has to choose between the survival of her son and an -unknown to her- tribe of 1.000 people. She chooses her son. Does this make her evil? Or non-good? She just let 1.000 people die to save 1 person, nevertheless many people might argue that their own mother might make the same choice.
I'm amazed that you've decided to be critical of alignment and yet seem to lack even the most basic understanding of it. You propose a very quintessential choice between the needs of the many and the needs of the one, between society and the individual, between the personal and the impersonal, and you seem to think that this is within the constrains of the alignment system principally a question of good and evil.
According to the alignment system, we certainly can't know whether this mother is good or evil (or even neutral). But we can know that she certainly isn't very piously lawful.
Is a person with schizophrenia evil? He might kill a dozen innocent men next morning for no reason, or be the most considerate, gentle, nice person to everyone for the next month.
First of all, schizophrenia is a separate disorder from multiple personality disorder. Schizophrenia is generally regarded as CN on the grounds it creates a private internal and unreviewable reality for the person. Sociopathic behavior on the other hand is generally regarded as NE in that it believes in the value of evil and destruction for its own sake. Many other personality disorders might simply be extreme personality quirks without forcing a particular alignment save where they force particular sorts of actions. The problem of multiple personality disorder is generally dealt with by giving each personality a separate alignment.
Why do we need to put a tab on a character's way of thinking and say "He is CN or NE"?
That's the heart of it. People who condemn the alignment system fundamentally do not wish the player character's thinking to be reviewable or labeled.
Secondly, mentalities are changeable. Past experiences shape the way of thinking. A character might begin NG, see cruelty in life and turn CN and then meet and be part of a kind family and turn CG or a totally different course that goes from LG to CE and back. It is still the same person. Only last time he adventured, anti-paladins could smite him and this time paladins can smite him.
Sure. And alignment is changeable as well. But it's quite possible to make a journey of personality change and character growth entirely within an alignment. For example, a character might be NG, see cruelty in life and become cold and cynical and perhaps pursue isolation and an acetic lifestyle, then meet and be part of a kind family and repent and become more warm and sociable, all without leaving his essential NG outlook on life.
Thirdly, there is a fine line between thinking of doing sth and actually doing it. A character wants to commit a very evil act. Nevertheless, he never does it. Was it because he never got the chance? Was it because sth internal stopped him every time? Only he knows (and sometimes not even him). Is he evil already? Does he become evil the moment he does it? How does a game base its mechanics on such a fine line that even the player might not be able to interpret?
This is precisely what alignment answers. The vast majority of evil characters in my campaign have never murdered anyone. What the evil (or good) alignment marker indicates for the DM whether, if they felt that murder advanced their interests, they would do so.
And lastly, there is the local perception of good / evil and law / chaos. Different mindsets might be considered evil somewhere or good somewhere else.
Sure. The people who are LE don't necessarily believe that they are in the wrong for being lawful evil. They may in fact believe that they are in the right, and that what you call 'evil' is in fact good. People who are piously CN don't believe that they are wrong - but that they are in the right. They see the fundamental root of all evil as being LN. LN's in their turn see the reverse. And even within an alignment there can be sharp disagreements. Two Lawful societies can have competing rules and interests. They may admire and understand the other, but still see each other as enemies. The personal interests of a Chaotic can of course sharply disagree with the personal interests of another Chaotic.
What the alignment system tells us though is that these things - evil and good - are real and not merely artificial constructs. Of course, even within that reality different alignments might believe that each alignment is less real than the other. For example, a NG person may believe that Good is a real and absolute thing, and that Evil (though real) is simply the absence of good. The same person may believe that Law and Chaos are artificial constructs of society and persons, and that - while the represent real traits - they are simply the flawed and broken shards of true understanding. And so forth for each of the alignments - the NE person may deny that good exists and claim instead that there are only different expressions of evil and that the 'righteous' are no better than anyone else.
Alignment allows us to resolve the complexities of relativism.
For me, the real heart of the matter is that I've never once in 30 years seen the removal of alignment lead to greater consideration of moral and ethical matters. The pretext that in doing so you are allowing for more nuanced and mature exploration in practice is always just a pretext. It invariably leads to things like, "We want to run an evil game, but we don't want to label or characters evil.", or has a basis like, "We are all in real life unreflective moral relativists and we don't want anything in our game that might challenge our understanding and make us think. We want a fundamentally chaotic game were each person is the sole judge of whether they are doing 'the right thing' or not." And so forth.
So yeah, I've got a bit of a bias against calls for removing alignment. I find them based on weak understanding of what alignment is and to generally have ulterior motives.