I can sympathise with the 'side-game' position because I held it for quite a long time as well, but Fynn is right. Pretty much all the games are a sizeable piece of the puzzle in the overall KH arc. Though if I had to judge which games were utterly essential to understand III, at a bare minimum I'd go I > CoM? > II > BBS > All of 2.8 > III. Though you might as well get the collections and play/watch all of them.

I played II before CoM, and the main thing that gets fleshed out is Riku. The core stuff with Sora, Namine and the Organisation is picked up in II well enough, but much of the Riku plot is confusing without CoM, so I do see that now. I still struggle with CoM's card system (especially with card duels), but you know. I just play on Beginner.

358/2 Days is also very important in fleshing out the relationships seen in II. You don't have to play 358/2 Days, but it really aids understanding for II and as a stand-alone entry, it is very good story. you lose nothing watching the movie version on 1.5. From what I hear, the actual gameplay of Days isn't that great.

BBS is probably more essential to the main arc. It is surprisingly long though. I kinda got fatigued part-way through Ventus's story (having completed Terra) and I kinda gave up. I'll get back to it. It can get slightly repetitive going through the worlds with another character (who you have to build from scratch again).

I also haven't played DDD either, but from what I hear that's also tied strongly to the main arc and sets up III. Pretty much all of the 2.8 set is set-up for III. When the price gets down, I'll grab 2.8 after completing BBS. It isn't as if III is getting anywhere fast