A CASE FOR RANDOM ROLL...
It's not just about stats, but what those stats mean...what they "say" about the character.
If you allow your players to rearrange their random rolls...
I do allow that in my current game, but that's the rules. So I play by 'em.
If I were running a 1E game, my preference is to be hard core. I'd probably go with the default 1E character generation rule.
....roll up characters away from the table....
I'd never allow that.
...or create new characters if they are unhappy with their stats...
Nope. What you roll is what you get.
Here's a real life example of a character in my game. He was created, by RAW (Conan RPG 2E rules), with no house rules. The player rolled stats exactly one time, in front of me, using 4d6, drop lowest, arrange to taste (which is the default Conan system).
After Racial modifiers were applied (1st level Cimmerian Barbarian), the character looks like this:
STR 19 (+4)
DEX 13 (+1)
CON 10 (+0)
INT 12 (+1)
WIS 7 (-2)
CHR 6 (-2)
A lot of players wouldn't be happy with this guy. He's strong, yes (rolled 17 with +2 racial STR), but many would see him as a one trick pony and a lot of negatives.
In my game, it doesn't matter. You get what you roll. My players are used to that.
So, we looked at this guy...what do his stats say about him?
First, he's unbelieveably strong. In a barbaric warrior culture, that's a good thing. He's better than average in DEX, and his CON is average. He'll make a good warrior for the clan.
But, he's got a problem. He's lacking in common sense. WIS 7. But, he's a cut above average in intelligence. INT 12. And, his personal charisma is quite low. CHA 6.
So, what does this say?
It says that he's shy, introverted, maybe a bit wierd, different, self conscious, and socially unacceptible on some counts. Maybe he's bitter--a real downer to be around.
This whole Charisma and Wisdom thing led to some brainstorming, and here's what we came up with...
Caelis is one of those warriors that you trust with your life to have your back, but you'd never invite him to your formal tea party or to meet your snooty friends (modern day example--think in Barbaric Cimmerian terms).
People meet him and don't like him. He's always putting his foot in his mouth.
To translate this to the Barbaric culture of the Cimmerians, we made up a pretty cool story:
Caelis' mother was pregnant. These barbarian had no idea that she was carrying triplets. When she gave birth, first Caels' brother was born (Branoc), and then Caelis. They found a third babe in the womb that was stillborne.
Well, these superstitious barbarians thought that Caelis and fought his brother (the dead one) and killed him in his mother's womb.
As Caelis grew up, villagers would give him the evil eye, stay away from him, treat him differently. This affected his social skills.
They attached a moniker to him: Redbirth. They call him Caelis Redbirth.
They fear him for his strength, but they also think that there is something unnatural about him.
Now, that's a pretty cool background story for a barbarian character, and it reasons why his CHR and WIS are so low.
This is why I love random character generation. It leads to this type of creativeness. I guarrantee you the character would not be as interesting (and he certainly wouldn't have those stats) if he were generated using point buy.
With point buy, we'd have a Caelis the hero, not Caelis Redbirth.
As far as your the Cons you cite against Random Generation:
Some characters will be much better than others.
Yes. That's like life. Why would you even want all your PCs equal? Is this a superhero group, or a group of adventurers?
Are Conan's companions as strong as Conan? Are Frodo and Sam, Aragorn, Legolas, Gimli, and Gandolf all balanced characters?
Diversity is interesting.
Because there is often a disparity, percieved or real, between the guy who has the 'cool character' and the guy with the bland or subpar one, it tends to greatly tempt players into cheating on their die rolls.
First, if I ever caught a player "cheating", I'd throw him out of my game.
Second, all characters can be "cool", and in my games, it's usually the weaker ones that are "cooler" because we've gotten creative to explain the character's stat weakness (as with Caelis Redbirth, above).
I have tons of examples of characters from my games with weak stats that became the "star" of the campaign.