I think the best case scenario for this team, as currently constructed, is a legitimate championship contender - I'm not going to go as far as to say they simply will win championships because for that even the best teams require some luck. There have been plenty of amazing, championship quality teams who simply haven't won chips because of bad luck, or running into another championship calibre team that gets the coinflip over them. Probably the best example of this in recent history was Golden State losing to the Cavs: obviously a championship level team, they literally won championships on the year before and after that season with the same core for the most part, but Lebron had one of the best playoff runs ever, Kyrie hit the shot, etc etc.
As for the specific developments this team would need to see in order to reach that level, I largely agree with your points.
1) Zion's defense would have to come back. I'm still one of those people who will die on the hill that his defense wasn't quite as bad as some people said the last time he played, but that doesn't mean I think it was good, because it wasn't. It was a negative, and clearly a weakness in his game. In order for him to be an MVP I don't really think defensive improvement is needed because we know the NBA will happily give MVP to players who don't play defense, but in order for him to peak, he will need that improvement. Early Draymond is a pretty good comparison in terms of where he could be on that end.
2) Herb takes a step up as a shooter. I think his defense is already all-league, and so there's no specific need for improvement there, and I don't think it's necessary for him to become a legitimate ballhandler, but he needs to go from being a meh shooter to a good one. If he can shoot 37 or 38% from 3 on reasonable volume (3.5 a game, maybe?) then that suddenly becomes very significant for his two-way impact. Obviously his game is different, but I think his mixture of defensive versatility and shooting with plus passing puts him in a similar category of impact and role as a 2006-2009 Kirilenko.
3) Trey needs to peak as a legitimately fantastic shooter. I think that's fairly likely in real life, but I'm talking something like 40% on 8 3PAs per 100 or more; in many ways he would be playing the Klay Thompson role for us in that he's capable of using size and length to maximise shooting impact and also cover perimeter defense, but I think Trey's athleticism is higher than Klay's and he has more on-ball ability. That's not to say he'll have a better career than Klay of course, but I think if he peaks he can be a more versatile threat, so 2015-16 Klay with plus ballhandling makes sense as a comparison.
4) CJ and BI are, I think, already pretty solid. What I would want from BI is for his shooting to return to his own form from a season ago. Last year his 3pt shooting dropped off in volume and percentage and he had some spells where he shot pretty terrible from behind the arc; not just dry spells in terms of percentage, but periods where he was refusing to take them as well. I think with Zion back we'll see those numbers come back up, and I think that's necessary for the team to peak as well. Not only does it make BI a more efficient threat but it obviously impacts spacing to a huge degree.
I think your Kings comparison is reasonable, but I would also like to throw my hat in the ring for something like the mid-2000s Suns. Now, it's tempting to think that's not the case because we're probably a slower team in today's NBA (not glacial, but not top 5 in pace) and the Suns have a reputation for that speedy offense, but in 2004-5 the Suns had a 95.9 Pace, which in today's NBA would be a bottom 5 pace. The game is way faster now, and I think that level of pace is fairly standard today. That year, the Suns won 62 games, led the league in rebounding, assists, and points per game. They were 2nd in the NBA in 2pt%, and led the league in ORtg while being fairly middle of the pack on DRtg.
Stylistically, I don't think they're a good comparison: we don't have Nash-like central hubs, our lead scoring big (Zion) is a more on-ball threat than Stoudemire, etc etc. But in terms of the on court result: a strong rebounding team that dominates offensively andd only has to avoid disaster on defense to peak, who can mow through the regular season by being super efficient from 2 and sharing the ball, while playing at probably around a 96-98 pace? That, I think, is a semi reasonable outcome to expect.
The downside is that the Suns never won a championship, and even in the season I mention they lost in the WCF. But while obviously getting that luck and winning a championship would be awesome, is there any Pels fan who would turn down what the Suns had then? They had 6 straight .500+ seasons, including seasons of 62, 54, 61, 55, and 54 wins, made the conference finals 3 times, and produced 2 MVP seasons for the lead star. Sure, a Championship is the peak but getting that would be legitimately the best period in Pelicans history by a huge margin, and I'd take it every day.