I'm in the weird state of being one of the few English speakers who runs/plays Japanese RPGs. I even translated and published some, too ("Tenra Bansho Zero", "Ryuutama", "Maid"). Plus I've got buddies in a few RPG/game companies over here in Japan.
So what you're looking for is unequivocally one of these few games that meet your requirements:
1) Sword World 2.0, the "D&D of Japan". It meets all your requirements save the "small number of stats", but since it's the basic D&D stats, it shouldn't be a problem.
The 2.0 version is a great intro game to the hobby. It's simple, small and cheap (about $12 USD for a hand-sized book), "Rulebook 1" of 3, but you do NOT need the other two to play, the other two have higher level abilities and extra side-classes/races, absolutely unneeded (and IMO not as solid as the core material) to play.
Tight rules, simple setting. 2d6 based.
But the best part is that as one of the pillars of the hobby over here, it's got a lot of online play and commentary. If you go on niconico and search for Sword World Replay (ソードワールド リプレイ or SW2.0 リプレイ) you'll find a lot of "real play" videos. Unfortunately I can't direct-link because my StackExchange cred is too low. I'll maybe post some links later to my Japanese RPG blog over at j-rpg dot com.
But basically, this game is one of the core games that Japanese gamers cut their teeth on, come in to the hobby to play, etc.
Re Sim vs Story, unfortunately there are VERY few games in Japan that are as Nar/Story-heavy in Japan as you'll find in the US. Even the indie culture over there focuses more on making more indie light sim RPGs than gooey hippie stuff (that I love!) over here like PTA, Apoc World, etc. Oh, before I forget, there's a Japanese version of FIASCO in the works, should be done within a year.
Having said that, though, SW2.0 is light enough sim that there's a TON of room for that "fruitful void" of exploration and stories outside of combat and abilities. It's not crunchy to death, and I've seen entire sessions happen with dice barely hitting the table (and when they do, they don't bog the story down).
2)Shinobigami (it's a very unique RPG in that the players make the story together, but it meets all your other requirements, and is very easy to understand and play). Also Blood Crusade is similar, but instead of modern fantasy/ninja, it's Castlevania style "all players against a monster" game.
Shinobigami (which will be coming out next year in English!) probably most closely meets your requirements, and it is indeed super popular in Japanese RPG circles... and while I actually love it more than SW, I think that SW is a little more representative of the hobby. In any case, it's brilliant: 2d6 system, 2-6 players plug GM, 3-4 hours in a one-shot and you're done. Simple setting ("World of Darkness but with Ninjas instead of Vampires"), and the book is divided into two parts: The first half is a complete replay with notes so you can see a game being played out. Then, when you're done, the rules section begins so that you can do the awesome thing you just read in the first half of the book. This is super unique in Japanese RPGs, and I love it.
Anyway, Shinobigami (シノビガミ) is also about $12-14 USD for a hand-sized well-organized book, only the core book needed, there are several supplements but they are totally unneeded.
In terms of Story/Nar vs Sim, I describe Shinobigami as "It's part Fiasco with Ninjas, part German Board Game". There's no super simmy parts, but there is combat. And combat exists in this cool German style board game milieu of resource management, blind information and bluffing. So it's not about "rolling to attack, rolling damage", it's more about using your abilities to get around what you think are the enemies defenses. Plus, the game runs equal parts co-op and LIGHT competitive play: You are not all out for each others' blood all the time, but depending on the scenario you may either be working together (but pushing each other down a little because you each have unique, conflicting goals) against a greater evil... or you could set it up as a free-for-all but where the ninjas all have complicated connections/emotions with each other.
It practically runs itself. And it's SUPER story heavy. In fact, it's the second of a line of games in the company's "Dice Fiction" line, a group of simple one-book games where the focus is on one-shots over campaigns, quickly reading and playing (in just a few minutes) rather than reading tomes and prepping for hours.
I've run it for hundreds of people (from everyone to "never played an RPG before" to "Ken Hite and other designers at conventions"), and I've never, ever had a bad experience with it. It is a game that turns dice rolls and player motivation into Story Sausage.
While you can run SW2.0 above in variable session lengths (1-2 hours not out of the question), Shinobigami takes a solid 3 at least (though a total one-shot focus), unless you are running a One GM Two Player session, then you might get it down to 2 hours. But that time flies, and is very memorable.
3) If you can still find a copy, Endbreakers is also an incredible game: SUPER SUPER SUPER simple and light (the core rules are about 2 pages). Like 3 stats, lots of sample art to get you motivated and into what's happening, unique high-fantasy anime-y world with simple descriptors.
However, the above are all book/print only. There is no Japanese market for RPG PDFs (just print only), and likely never will be for the established publishers.
If you want a PDF game that's excellent, though, my friend Okada made his game publicly available in Japanese, for free even, in PDF format. Literally one of the only games in this format. It's called "Ryuutama", it's best described as a "Heartwarming travel fantasy RPG". You can download it for free LEGALLY here, you just need to create an account:
As noted, it requires all polyhedrals (the author made a love-letter to the unique gaming dice in the TRPG hobby), but you can get by with a $2.00 Mach Dice app or similar on a smartphone. The game is a novice-focused game about travel and wonder. You draft up simple characters that would be the "NPCs" of another fantasy world (the character classes are not Warrior, Wizard, etc; but rather Healer, Minstrel, Artisan, etc). Everyone in the setting eventually gets wanderlust, and thus team up with strangers and explore the world.
In Ryuutama, the players assist with scenario creation, everyone contributes an element to scenarios/towns and even the world/setting, so that every player has a draw into the shared world. Finally, the GM also has a character with their own character sheet, a dragon-person who watches over and protects the players (they can use limited powers to help the players reroll dice, rewind time, etc). Simple and fun.
Note, I am also publishing Ryuutama in English. It's available for Preorder now (PDF will be ready soon, book in a few months). We had a successful run at Kickstarter recently.
Note: This game is also available currently in print in French, and within the next 1-2 years in Spanish, Portuguese and German.
I hope this all helps. Let me know if you have any questions! I put a bunch of links to my online presence in my profile, so feel free to contact me offsite as well.