The game is balanced around asome given character power level, and varying stats GREATLY affect this balance.
While the general idea of rolling randomly has some appeal, you have to remember that this balance discrepancy is going to be a CAMPAIGN LONG aspect.
Think if, game #1, you allowed each player to pick up a green d4, or a red d4, then they all roll all at once. Then you say: okay, those that rolled a green die, every weapon you wielld gains that die number as a "non-magical innate divine blessing", which is a bonus to every action based or DC based on your main stat, until end of campaign. But for those that rolled a red die, it is the erxact opposite, a non-magical unremovable permanent demonic curse to all their main stat actions and DC, untill end of campaign.
Then watch the "unfair!" whinings and tantrums blow out from the red die players, and the players dropout. Eventuallly this repeats with new players, until you have EVERYBOPY with a green-die PC remaining.
Over 30 years of DMing, my personal expewrience is that 99% of players WNATING to roll, they don't REALLY want to have the randomness. At allo. what they really want is the opportunity to get a "more powerful" PC. Ideallly, more powerful than the other PCs. the rolled low? They wwhne, complain, ask to roll again, invent some excuses the very next game thhat "they don't oike that class and want to roll a new PC", or even outright SUICIDE their PC to make sure they get to roll a new one. O
In non too obvbious ways of course, such as taking extremely angerous ruisdks, while not fully healed, covering it with a bit of dramatic roleplay and "background personality reasons". Oh my PC is very, very proud, and also protective of X (other PC he "likes"), so he won't accept to get healed only for a few measily scratches (despite being only at 20% HP he just says "Oooh, I'm fiiiiine!" saying wen he inevitably die "
That was just my PC being too proud to admit he was badly hurt, you should have done a medical check to find ouyt how he REALLLY was hurt if you really wanted to heall him up!"), saying he perfers the healers keep their healing magic available to cure X, "in case it's needed".
Then playing the reckless imppatient self-sacrificing hero that tries to solve everything by taking on the most risks. "That corridor is full of traps but hey, we have enemies around, disarming all those traps would take way too long, SO I RUN ALONG THE CORRIDOR TO TRIGGER ALL THE TRAPS!"
Of course they typically die before the end of the 1st session.
Then it's Lather rinse Repeat until they finally get that uber-stats PC.
If prevented to do any of thatm, they tend to simply drop out of the campaign.
Thus clearly proving THEY WERE NOT AT ALL INTERESTED IN THEIR STATS BEING 'RANDOM', BUT IN THEM BEING 'MOAR POWARFULL FOR FREE'.
It's almost always done as a form of desire to get some free extra power creep. NOT as a way to get "variability".
I use Default Array and any new player STRONGLY insisting to roll, is just not reinvited. It always leads to way more problems than it's worth.