here's how I think magic should be, since "randomly generated" is clearly not specific enough
Yeah you noticed that a lot of people just say "Random" instead of actually saying what random is a lot of the time didn't you?
Anyhow magic cannot exactly work the way you would like. While mechanical effects are lovely you also need a hint of majesty and profane. Those are the kinds of things that are going to need more dirrect creation as the game wouldn't be able to tell the difference between a spell that causes someone to die and a spell that heals someone.
A lot of what makes magic what it is fits entirely within the imagry that it inspires. Necromancy and generic holy magic are extremely similar.
the player should be at the mercy of the random number generator when it comes to obtaining magic
I could understand it being tough, but remember that one of the things you do not want to create is a "character creation gauntlet". If magic is so rare that just to have it you have to be one of the "lucky ones" then simply allow an option to generate as one of the lucky ones.
I mean you generate as a Demigod as it is and in the future they will have genuin powers (as right now... Demigod is more accurately an accelerated start).
"when you find an wand of fire in nethack or D&D, it's "magic" in the sense that it defies science, but there's nothing wonderous or special or mysterious about it."
Actually within the setting of dungeons and dragons magic is a external source of energy that has been studied for centuries and is in fact intigrated into science. In fact the major difference between a Sorcerer and a Wizard is that a Sorcerer's powers comes from a natural magical gift and thus his magic is a natural extension of himself while the Wizard's magic comes from years of study and mental preperation to manipulate the magic around him. Hense why a Sorcerer uses Charisma and a Wizard uses intelligence.
Very few people within Dungeons and Dragons can even hope to manipulate the magical forces at any conceivable scale and even of those who have the ability, very few have enough talent to take it anywhere. To put it in perspective a Fireball requires a genius and would require I believe 4 ENTIRE pages in a large tome to even diagram how to properly cast it.
Its mundane appearance comes in two forms. One is the mechanics which cuts out all the dangerous and boring research a wizard would actually have to do just to know magic and glosses over the sacrifice they have to do to even create magic items, the second is that players are free to learn magic at any point and have little problem doing so.
You have to think about what exactly are you expecting the player to do.
Also if the player is so knowledgable of their own world that they can predict EXACTLY what it is they need... then obviously it is their benefit.
If they went to a Griffon who was bestowed the light of the gods as another character... and used that knowledge for another character to also learn that ability (somehow) then good for them.
You forget that with too much uncertainty you get goo... Just lifeless Goo.
Magic without logic is just whirling chaos.
Opportunities to get magic should be out of the player's contro
No by all means it should be within the player's ability or else it is less "Magic" and more double blind lottery. Spontaniously learning magic though, I can agree with. Fiction is brimmed with people who learned magic simply by enjoying nature or sparks of inspiration (I miss that book... though from what I understand it gets less interesting as it goes on).
Guarenteed magic, yes that is out of the player's control. Doing things that has a good possibility in ending with magic sounds perfectly logical (if he even knows what it is)
From a logical standpoint how would "Player has no control" even make sense unless there were even MORE specific barriers that head in the exact opposite dirrection your indicating (which is that magic should be rare and spontanious... however to take it out of the player's control makes it very specific)