It's been forever since I watched the first .hack series, but my personal recollection was that it didn't really do the concept of "This is a game!" much justice, either. Yes, we had Tsukasa "stuck" and all the oddities related to Aura and the World, but if we're being honest, you could probably revise Tsukasa's character to simply being some kind of amnesiac mage and you wouldn't see a whole lot of difference in the plot outside of occasional IRL references with Bear, Rose, and such.
SAO suffers somewhat similarly, but I'd also say they tried to be a bit more MMO about it. Yes, the full-dive tech allows for more feasibility in getting stuck, but it also enables a higher degree of awareness and control. As established early on between Klein and Kirito, using skills wasn't so much a matter of pushing the right buttons as it was merely thinking about using them with the appropriate conditions met like range or MP. SAO skill development also didn't seem so rigidly class-locked, as demonstrated by how Kirito went about learning dual-wield or countering the lower leveled PKers with significant regen. You also had craft skills that leveled up similarly to how we might know here with XIV or XI. Basically, the game liked to reward dedicated use of abilities or combinations thereof. And while their hand wavey menu management might not have seemed like the most detailed means of window browsing, I'm more inclined to say that was intentionally kept minimal for entertainment value purposes.
As for Log Horizon, I'll agree it tried the most to be MMO-like, but I'd almost say there was a conceptual equivalent of the uncanny valley that went on. The raids rarely felt like anything I'd personally experienced when gaming, and I don't think that's just a JP vs. NA thing. Meanwhile, they went to great lengths to try and convey a realistic economy and/or the politics behind such, but too many MMOs nowadays more or less treat crafting as an afterthought with the best gear coming from raids/tokens. Things like RMT or people maliciously manipulating markets don't really come off as a severe problem for the not-poor, which Shiro and his immediate circle of friends obviously were not starting as high level raiders. Watching the lowbie group also felt particularly painful at times, but also became an unfortunate requirement to the series precisely because of Shiro's starting status. On the other hand, I don't think it's fair for me to be passing a final judgment on this because it's a show that seems indefinitely incomplete.
Either way, saying this genre rarely feels like it gets it right has an incredibly highly diverse list of characteristics behind the sentiment. You've got the woes of the newbie in figuring out the basics. You've then got the mid-game building upon those with ideally increasing challenges, often requiring interactions with others. PK aside, you've got things like job/gear snobbery, disagreement on tactics, those of varying skill levels/receptivity, ninja lotting, carrying their weight or seriousness in play, actually finding people with matching agendas when you want to do something, sensible expectations of play time, and then some. Endgame just tends to take all that and dials it up to 11, where grinds are stupid like XI's RMEs or worthwhile solo content pretty much ceases to exist because raider feefees. Relative to the DoL show, a lot of this was either skimmed or was never really a factor at all.
Essentially, you rarely get the vibe players are individually selfish, and not to be mean, but to prioritize personal progress. They're either happily helpful or want to kill you (in the game and real life!). There's not a whole lot of in-between that goes on. Early/mid-SAO Kirito is probably how I'd best associate myself with characters in the genre as when there is a game I latch onto, I tend to accrue as much knowledge and power as I can on my own. The identity of coming off as "godly" may not seem as prevalent in reality nowadays or like early NA launch XI, but there are still people who may come to appreciate that should I chance upon helping them out (and level caps aren't a thing--boo hiss).
Violence good. **** bad. Yay America.