I absolutely agree with Celebrims way early beginning points about the flaws of "rolling" compared to point buy. But I still hate point buy and I still like my random method.
I'm not completely happy with either method. I'm just convinced random doesn't work and makes for a worse game. My problems with point buy have more to do with how players choose to play though than anything inherently wrong with it.
Because the rolling method gives me a challenge to work with and inspiration.
Rolling up characters is fun, but its not very functional for PCs (and non-functional for NPCs as well if it deprotagonizes the PCs). The fun of rolling up characters - not knowing what you are going to get, in a nutshell - is more than counterbalanced by the fact that usually someone is going to get what they don't want, the method isn't fair, the fact that it makes it harder to give advice on how to balance scenarios to a group's abilities, and the pressure it puts on the metagame. The better someone else's results, the less satisfactory your own results look. The better results you've had in the past, the more disappointing a bad roll is. The more difficult it is to engage the main game through your character's abilities, the more likely you are to be tempted to make your own fun and play a different game than everyone else is playing. And so forth.
I feel the justifications for that tend to be pretty weak.
Balance is not an inherent property of randomness. Randomness produces an arbitrary result. Fairness requires that players get about the same thing. Justice requires that they get what they deserve. Randomness misses both marks. This isn't a game show. We aren't randomly selecting winners and contestants. We are playing a game together.
Diversity is not an inherent property of randomness. We expect if we flip 5 heads in a row the next coin is less likely to be heads, but that's not true. If Diversity is your goal, you can guarantee diversity simply by choosing not being like the last X characters.
If inspiration is what you need because you are suffering from choice overload, there are plenty of ways to manage that that don't end up with the lack of balance of true randomness. A functional random chargen would be randomly picking between things that are balanced. For example, randomly selecting a class and race (assuming those are balance). Or randomly selecting a highest ability score, then randomly selecting your second highest, and so forth. Or you could randomly order your stats, generate 5 stats in order, and then buy the sixth one with your remaining points. Or you could roll up random stats and then match them as closely as possible using point buy. This wouldn't be purely random, but has the aspect of the producing the unknown and would generally not have strong balance problems in the outcomes. So people get the thrill of not knowing what they are going to get, and they get the inspiration to play something that they might not have considered, but they don't break the Fundamental Law of Roleplaying.
Gambling is an inherent property of random levels of resources, but not I think a functional one in the context of a cooperative or competive game. Winning is fun, but losing is not. Gamblers gamble to win. If they lose, this puts even more pressure on the metagame - chargen is becoming a competition. The person motivated primarily by the thrill of gambling is going to compulsively gamble trying to win 'the jackpot' - I lost, but I'll win the next time. What that tends to mean is you are setting your balance at the jackpot, and then going through frustration before getting what you actually want - the win. And I think that some people who say that they like random, are probably actually saying that they don't like point buy for some reason or the other. For example, if they've been used to playing 34 point buy characters (or higher!) using some random generation method (Method 1 with multiple rerolls, Method 3, Method 5, cheating), setting point buy to 25 feels like a rip off beyond any other aspects of rolling you miss. If point buy limits you beyond what you are used to getting from 'random', then it feels like point buy is saying, "You can never win."