To me, the AR became popular because of a demographic/cultural shift as much as anything, and not because of the rifle itself.
"Gun culture 1.0", you could call it (let's spitball it as 1960-2000) tended to see guns as more of tools than "shooters" to blast away with. Almost everyone had guns, but they were more of a means to an end. You had a rifle or shotgun for hunting, or a .38 in the drawer for home defense. A guy who was "into guns" was someone who spent a bunch of money on some nice Weatherby magnums and Browning shotguns. Not a lot of people went to the range just to target shoot for the heck of it. There were always hobbyists and plinking got more common as time went on, but plinkers specifically will always shoot what's cheap. With all the cheap surplus guns on the market - depends on the specific time, but Springfields, M1 Carbines, Mosins, SKSs, and MAK 90s could all be had for next to nothing at different points - there were LOTS of options besides an AR.
ARs at this time were either very expensive or had some really dubious quality, so they didn't fit the mold of plinkers, and at the time they just weren't hunting rifles. I never saw an AR varmint rig till about 2000ish, and the AR wasn't suited any kind of big game because there just weren't the calibers for it. The first big bore AR cartridge I can personally remember hearing about was the 50 Beowulf, and that wasn't till Bush Jr's first term. So buying an M4 type rifle like a Colt 6920 for like $1000 at a time around say the 80s when SKSs were $80 and Minis were like $300 raised some eyebrows. You're clearly not going hunting with that thing, and it's way expensive/non mainstream for a fun gun... so what are you planning to do with it? Militia-wacko type stuff wasn't an illogical conclusion. It sounds Fuddish to us now, but it was a very Fuddish gun culture and there were reasons for it.
Then over the last generation, you start seeing a shift in America towards an more and more urban environment, and that's shifted people away from hunting and more towards a "gun culture 2.0" centered around shooting. The population of the US has gone from 180 million in 1960, to 250 million in 1990, to about 330 million today, and the vast majority of that growth has been around cities. More Americans are living there, and more and more open land for hunting and shooting is getting chewed up as our suburbs continue to bloat outwards. My dad's favorite hunting spot for grouse in the 1970s is now a four-lane highway with a Target at the exit. So the role of guns has changed, as there's simply nowhere for many people to shoot. For a lot of people, the only place they can go now is the range, where some of the good old boys in the Mountain West 50 years ago had never even seen a rifle range. And range shooting entails a whole different set of parameters than hunting, especially if the distance caps out at 100 yards or so like a lot of places. Trajectory and terminal effect stop mattering, so put away the 300 Win Mags, and accuracy and cheap ammo become by far the biggest priorities.
Around the time of the AWB, more people started buying ARs as a take-that to the govt, fear of further gun control, etc, and people started realizing that they're actually some pretty freaking sweet shooters. Even something like a Olympic will shoot circles around an SKS or a Mini, and you can feed an AR for $7 a box of ammo, and so ARs were just a perfect fit for what your average city-dweller shooter was looking to do. I'd say ARs were starting to become mainstream by about 2008. And as that happened, prices were driven down, quality was driven up as some of the incredibly chintzy 90s crap like cast receivers and recycled Vietnam M16 takeoff parts was well past not flying any more, and that's ultimately what's brought us to today. If you want the best gun to shoot on a range, it's the AR. If you're a casual plinker and want the cheapest gun to shoot on the range, it's an AR. Those kinds of people buy $300 PSA kits now the same way they used to buy $80 SKSs out of Shotgun News in the 90s. ARs just totally dominate the market and between cost, quality, and need, I just don't see that changing soon.