This fanbase has a weird tendency to try to attach an asterisk to our own title run. Leave that nonsense for bitter rivals.
My thing is, neither guy is perfect. Ime and Joe. This thread is proving it too with all the back and forths on the flaws and shortcomings from each guy. And it's all valid.
It's not like we're debating Popovich vs. Joe Mazzulla lol. Ime is a great coach too. He's not perfect. Joe isn't either.
Personally at some point it's just down to personal preference. If you're a defensive minded guy who wants to see discipline in your team, Ime's the best coach for that. For someone who wants to see points getting racked up and a more zen-type, collaborative approach to coaching backed by analytics and nerds, they will like what Joe has done for the team. But if we won the championship despite Joe, and could have easily won it with Ime, then what we're really saying is the coach doesn't really have an influence on the fortunes to the team...so it doesn't really matter who it is, or whether they even have one for that matter.
I remember when I lived in Australia in 2000 and they were having a debate about how useful the coach of their (apparently) ATG cricket team was, one of the players said that a coach (bus) was the vehicle they traveled to and from the ground, the actual coach they had was less useful than that 
So if we can win in spite of whoever it is, because our team is good enough to overcome a bad coach, then ultimately it's all about the players and it doesn't really matter who it is, so we have nothing to worry about - we just need to make sure our players are on their game so they can coach themselves come playoff time... I feel like they're not as influential in basketball games as they are in football games for example, just the same as soccer coaches are useless once the game starts, it's all up to the players. They can do the occasional ATO or make the subs, that's about it. Maybe as Tenn said above, Joe just needs to make sure they keep playing hard for him, and leave the schemes and coverages to them 
Maybe a little extreme ?
Of course the coach matters. I liked your description of these two coaches, pretty accurate. My opinion though is that, yes, we won last year with a coach who had a couple of strengths that did help the team, but who overall has too many negatives to keep winning. I think that with a much better coach, this very good team could be so much stronger and could win multiple titles.
Fact of the matter is, can't just rely solely on 3's to win games. If those 3's aren't falling, then what other offense do you have. Need to get creative or trim down on this 3-ball philosophy and be more dynamic with offenses. And of course, getting the team to play defense is another challenge.
Right on target.
I'm sitting here watching the Tenn - Ky game tonite and find it interesting to see both teams running lots of off-ball movement & screening.
Sure doesn't appear all that difficult to do. For me, it is mind boggling that Mazzulla can't implement some of this into the Celtics offensive scheme. What's the harm ? It opens up plenty of 3-point opportunities if that's what he is worried about. It would be a travesty to get easier shots from, say, 10' in. We couldn't possibly win without all the dribble-heavy, fallaway, contested 3's we take early in the shot clock.
Ok. How many of these players are even going to sniff the G League? How many of these coaches are going to end up on an NBA staff in the next year or two? Let's count them on our hands.
Now, with your remaining nine-and-a-half fingers, let's type our reply.
If you want an example of how a university-level system hides how your talent will translate at the pro level, look at Jimmer Fredette. If you want to look at how this works at an NBA level, look at the guys that Brad Stevens coached during the
lean years.
Now, let's remember, Stevens is a good coach in the College Ball(TM) sense, right? Who hired Mazzulla? Who kept Mazzulla on as a head coach? How are we not connecting these very obvious dots?
In terms of what's the harm: the harm is quite simply that the coaching staff, the front office, and the players - who are paid millions of dollars to figure out the best ways to win basketball games - have decided that for this roster this is not the best way to win basketball games.
We're all free to disagree, but this persistent narrative that the front office, the coaching staff, and the players are somehow blind to the things that are
so obvious to us, the plebs watching the team play for (generously) an hour and a half per week is mindboggling. That's coaching agnostic, by the way. I've said the same thing regarding Ime, Stevens, Doc, etc. They all make mistakes, but mistakes are not strategy [which is not tactics, either] and shouldn't be treated as such.