Two things you should know before I start. Firstly, I love me some Star Wars, and have for over 2/3rds of my life now. Secondly, I am a gigantic military history nerd who occasionally reads theses from West Point's archive <insert Homer Nerrrd gif>. There's been a heck of a lot of debate over Poe Dameron's actions during The Last Jedi, and a not insubstantial amount of it has taken place in my head, and my take is that the conflict between Poe and Holdo represents the conflict between what the Resistance is, and what it thinks it is.
The Resistance consistently runs into problems in The Last Jedi because it still sees itself as the legitimate government of the galaxy, and behaves accordingly. A fascinating glimpse of this mentality is shown when they send out the distress call from Crait. I could hate the First Order with the passion of a burning supernova, but there would be no way in hell I would send my men and ships to back THAT losing horse.
Then again, this is business as usual for the Rebel Alliance. They fight by committee, if Return of the Jedi and Rogue One are anything to go by, and unlike the mono-species Empire, have to take the interests of vastly different groups into account. This is the only explanation for the bizarre decision in ROTJ to send the entire fleet to Endor. They weren't expecting the Imperial Star Destroyers, and there's no way in hell they're going to damage the Death Star, so why are they there? Strike force on the ground (which they do) and a handful of small fighters, and if the worst happens that's all you lose, and Ackbar can go wreak havoc elsewhere. In fact, this idea may be even better. Can you imagine the damage the Rebel Fleet could have inflicted if Gold Leader had sent a message saying "Just FYI, the Imperial Fleet's here too." after arriving at Endor?
The only explanation is that the Alliance is a loose confederation whose strategic initiatives have to conform to politics within the group, and I suspect that the Resistance is similar. The fact that they are evacuating a base under enemy fire at the start suggests that things have gone very, very wrong.
Forget anything that happens in the movie - the worst strategic decisions made have been made a while before the movie begins.
Poe's ill advised assault on the Dreadnought in these circumstances is understandable, but still idiotic. His insistence on destroying the ship shows that he simply does not understand that the Resistance is now in such bad shape that the Dreadnought is less valuable and more replaceable than literally anything that they would lose to take it out.
But with that being the case, there was no reason for the mission to be ordered in the first place. Either don't order the mission, or let it follow through. As it was, the Resistance command managed to get the worst of both worlds - relying on their known hothead pilot to pull a mission and refusing to, and then blame him for its Pyrrhic success. No wonder he's pissed, even if he should, I dunno, follow orders?
If Poe represents one end of the spectrum, then Holdo represents the other. She keeps her strategy secret from her senior command for seemingly very little reason other than, I dunno, security I guess? But the trouble is she's in a position where a lot of the senior command staff have been wiped out and they are under an extreme time constraint. Plus all of their fighter cover is gone.
So why are you freezing out Poe Dameron? He's a clever guy and while hotheaded he also shows a solid strategic grasp of the situation after the initial run on the Dreadnought. The fact that he plans and executes two separate attempts to save the fleet independently shows that he's an ideas guy you need on your side. One thing the Resistance and Rebels have been very good at is utilizing mid-level talent.
The problem is that Holdo is still acting like the Resistance is a legitimate military force that has suffered a setback, and that the chain of command still holds true. It's a severe flaw as a commander, an inflexibility in the face of drastically changed circumstances, and betrays the same mentality as the Crait distress call and Poe's attack on the Dreadnought.
Poe's attempt at a coup is borne out of a lack of information and represents a failure of Holdo as a commander, and the fact that he was semi-successful (i.e. a bunch of people supported him) lends credence to the idea that she completely failed to judge the situation.
Which, to be fair, was the highest possible-stress situation imaginable. Like I said: Worst mistakes -> before the movie.
Having said THAT Holdo's sacrifice is undoubtedly heroic (not to mention completely badass), and shows what the Resistance and the Rebel Alliance was really about: punching above their weight, which all successful insurgencies have to do.
Poe also accepts the other thing an insurgency has to do is run away a lot, because in order to fight, first you have to survive.
In the end, both their arcs come to the same point. Poe and Holdo have to learn to think like a Rebel.