So the main question I think has to be addressed to decide how you want to do it in your game, is where do you included choice, and why? Traditionally, you don't pick your ability scores, or you hit points, or the spells you learn on level up, but you do get to pick your race and class.
You do, though tables are provided for those who want to roll for species and-or class as well.
Further, the chooseable-species list is limited by where the party is in the setting. Humans are always chooseable, but after that you can only choose things that live "in the neighbourhood", so to speak. What this means is, for example, that if there's no Dwarvish lands within a thousand miles of where you'll be joining (or starting) the party, Dwarf isn't chooseable as your species - you have to roll for it and might very well not get it. But if you're close to Dwarven lands, Dwarf will be chooseable.
There's only a very few regions in my setting where all the major species would be chooseable.
With just those variables, I think race stands out as an odd one to pick. If you only got to pick your class, then there would be some consistency. Here is a random person you get, now make of them what you can. But why do you get to choose to be born an elf, but not necessarily a graceful one? It's a weird agency insertion point.
Interesting point.
Back when, species was something that had to be rolled for in our games - you rolled until you got two different options, then chose one of them. Over time, this slowly got relaxed.
I get nothing out of playing unequal characters, unless that's an explicitly presented and accepted part of the plan, and there is a reason that works for me. So if I'm presented with a situation where a party of characters is going to have vastly different roles in a narrative focus, like the Fellowship of the Ring, then I might be interested.
The Fellowship is, in some ways at least, exactly what I expect a typical D&D party to look like (though usually with not quite so many Hobbits
) once it gets a few levels in. Different abilities, different levels, yet with everyone still able to contirbute even though each character's end result might not be the same.
But if it's a game where the goal is to face challenges and grow in power, everyone not starting at the same level is a complete disincentive to my participation. I'm mostly referring to long-term play though. I'm game for just about anything for a one-shot lasting 4 or less sessions.
I'm only referring to long term play. And over the long term, things tend to change. A character who starts out seemingly very powerful in relation to its peers might hit rough times and end up the weakest in the party. Flip side: a character who started out without much going for it could end up a superstar (these are my favourites!).
And in my game, levels come and go - yes, level drain is a thing. Items and wealth come and go, and a good ol' meltdown can set you back a long way (and see below).
I can't recall the precise system I used for the equal random generation (and the next time I'll improve it), but I can remember bits.
-I had a number that the raw random ability scores eventually needed to add up to. The stats were intended to be high for campaign-specific reasons, so I believe it was something like 80 or 82.
-We started with everyone rolling six ability scores. It was probably 4d6 drop lowest.
-After that, there was a cycle of rolling a d6 to target one of the ability scores, and then (I think) rolling another d6 to add or subtract from it, based on whether the ability score total was above or below the target total value.
Something like that. It was a little too messy, but I only needed it once and it generated some interesting results. The point buy option was a more typical 5e point buy, with a few more points. Because there is an increasing cost with point buy, and you can't efficiently get stats very low, the way it worked out was if you wanted to keep your stats pretty close to each other (hovering around 14), your ability scores could add up to a higher total than the random method. But, the random method could let you start with higher values that you couldn't otherwise buy with point buy. This meant you wouldn't want to take one of the random values unless it was giving you a wide stat range, because you could otherwise do better buying stats. As I said, I consider it a success because I had people carefully consider it and take both options.
Sounds complex, but if it worked out for you, all is good!
The luck angle is an interesting idea. I'm sure there is some sort of deeper analysis that could go on here. The way I experience it, I want luck to mostly have a short-term effect, not a permanent one. Permanent effects are more desirable to me as the results of consistent choices over a long span of time. Sure, in the real world some bad luck can ruin your life, but I don't find that a particularly desirable thing to bring into my entertainment.
Several things here.
First, IME - and with lots of number-crunching to back it up - initial stat roll average isn't much of a predictor of how long a character's career will be.
Second, while a run of bad luck can ruin you the game probably provides more opportunity to bounce back than does real life. My currently-active Thief is proof of this - she came in at lower level than anyone else in the party and then, on first meeting the group, promptly lost all her magic to a Deck of Many Things (meanwhile, some other characters cleaned up!). That was 15 sessions ago or so, and she's still going, slowly re-equipping herself and finding ways to contribute wherever she can.
Third, while bad luck can ruin your life the flip side is that good luck - which also rears its head now and then - can springboard you to great things.
And I'm the kind of player who's willing to go through a lot more of a "grueling" play experience than anyone else I play with, so that's saying something for me.
But again, it's all about knowing what the play goal is, and figuring out efficient systems to provide it, rather than fighting systems that are getting in the way.
My preference is a system that just kinda sits there, and gets/doesn't get in the way about equally no matter what you try to do with it.
