I fail to see why random stats would preclude that.
In my current campaign we both rolled random stats in order and determined elements of our character's stories and personalities randomly.
It resulted in great and interesting characters.
Because I absolutely, utterly despise being
told what story to explore, especially by something without will or motive. I am quite open to suggestions from peers, in the extremely rare event that I don't have a concept already in mind. Suggestions are malleable clay, capable of being molded by other decisions and still leaving me as the principal author of the character.
Truly random stats, none of that wishy-washy "roll and assign" stuff (which is, now that I reread, precisely what you did--strict roll order, that is) deny me that agency. If I really wanted to play a caster, but have all mental scores in the 10-or-less region, I'm SOL. If I want to play an acrobatic Monk or Rogue and get an 8 Dex, I'm SOL. If I roll substantially outside the average of the rest of the party, for better or for worse, then I'm going to feel bad: if better, because my character will outdo and probably outplay the others noticeably, and I'll feel guilty despite having done nothing wrong; if worse, because I'll feel like dead weight holding the party back, and may feel resentful even though I know no one is "at fault" for the situation.
If I have a specific story I want to investigate, and I almost always do, I want to be able to determine it myself. I hate the idea of throwing myself before the randomness of "ability roulette." Give me point buy any day of the week.