Do you ALWAYS ALWAYS keep the first set you roll, or do you have some kind of "oh that's okay, roll again" when you inevitably roll crappy?
I find it interesting when people have '4d6, keep 3' rules, coupled with 'if you roll lower than 72, you can reroll'. Remember, 4d6 keep 3 is supposed to simulate being above average already; it's in comparison to a 'normal' - i.e. non-adventuring - person having 3d6 ability scores.
The expected total of six rolled ability scores is 73, so the reroll rule basically dictates that PCs are above average
for the above-average roll.If you're going to have a reroll rule, it ought to include a ceiling as well as a floor. Given that the intent is to be better than a 3d6 character, you could quite reasonably set the floor at 63, with a symmetrical ceiling at 83. Roll too high, and you need to reroll for worse abilities!
There are other ways to bound it, and other ways to avoid wild power discrepancies between characters when rolling - but really, if you don't want to take the stats you rolled, just don't roll stats.
4d6-L arranged to taste. If your stats don’t meet the standard array you can take it.
This is almost what I'd do for rolled stats in a campaign, except that players get to roll once, then pick between taht array and the standard array. It's possible someone wants to play a character that's seriously MAD, so would rather have three excellent rolls and three abysmal rolls than the standard array.
Now if you truly wanted to randomise the concept in 5e, much better way would be to choose your species, background, class and subclass at random.
One method I've suggested for 5e is to lock in your character origin (race/species/lineage, depending on the book), including any ASIs that come from it. Then roll 4d6 drop lowest in order, and lock that in. Then pick a class that sort of works, ish, with the character you've 'born' into the world.