re: What makes a great or Championship-level NBA team?

Ive wondered the same thing. I'm a casual basketball fan who just started getting into the Pelicans this season, mainly because my 15 yr. old son has taken a sudden interest in basketball. From attending the Spurs and Golden State games, it looks like we have the pieces to play with anyone when we're healthy and everyone is playing well.
At the Spurs game, despite his awesome stats, Davis never really looked "hot" (at least not like he did in the 3rd quarter of the GS game). But he stayed involved in the offense from start to finish, whereas in some other games he sometimes seemed to disappear for a little while, then turn it on in spurts.
The guard play looked great against the Suns. Jrue looked amazing while he was in. Reke was being Reke and killing it early on by driving the basket, plus hitting some other great shots. Cole hit the nice mid-range stuff, but looked great driving to the basket as well, which I hadn't seem from him in other games I've attended. Gordon never seemed to really heat up, but he hit some shots at crucial moments. While we don't have a superstar at guard, if all 4 guards are playing well it seems like they can play with just about anyone.
Now, from what I've read here, it's my understanding that Reke and especially Cole were able to drive the lane because the Spurs made a point of shutting down Anderson (and I would assume QPon too since neither ever got hot hitting a streak of 3s like in most games). That's great, and if they don't shut them down, when the guards can't light up the lane we just need Anderson and QPon to light it up from 3 pt. range.
And Asik just needs to be Asik.
When those things happen - in other words, when each player is just "doing what they do" - evidence shows that we can play with anyone.
That being said, I guess the importance of two superstars is that superstars are always "on."
I don't know what the missing pieces are, and would like to know what we need to get to the next level - just like the OP asked. But it sure seems like we can hang with anyone out there when we're healthy and firing on all cylinders.

ETA: Since I just bought season tickets for me and my son, I was warning him how those early season games are not going to be as exciting as these recent few since nothing really appears to be "on the line" (aka playoffs) that early on. Most of the handful of games we went to this year were near the end of the season. But I'm really looking forward to watching the team develop over the course of next season, and I have a real desire to learn about all the smaller pieces of the puzzle over a full season.