The action was fun, but no, it didn't influence the plot at all. In fact, Poe, Rose, and Finn's sideplot had literally no impact on anything.
Right because the importance of handling and learning from failure, and the meta-narrative of "doing Star Wars the way we have always done Star Wars will not work in the future" weren't, like, the major themes of the movie and reflected in these plots. The major philosophical debate was whether Star Wars (or, the Jedi) should continue to exist, with our protagonist setting it to build something new and our antagonist setting out to burn it all down (he even says "Let the past die... Kill it if you have to." Poe, as the flyboy whose decisions generally turn out well for the universe finds himself killing most of his squadron in an ill-conceived attack, and ruining am otherwise viable plan, killing way more of the Resistance in the process, because he didn't trust the women in charge. He failed and learned from it.
Finn failed in his mission and learned from it.
The introduction of Holdo as Leia's replacement, aside from being sudden, was made worse from the fact that she clearly had an axe to grind against the Heroes of the Rebellion.
Was she wrong, though? I thought the movie did an excellent job showing us she was right about a protagonist we had grown to love.
I like the idea that her perspective on Poe was that heroics can lead to needless sacrifice, but the way she demonstrated that was withholding information for literally no reason at all other than "ell oh ell, Poe is a dumb jock." He outright asks her if there is a destination, if they are headed somewhere or just running without direction. He is pleading with her to give him a reason to go on. Her response is "Leia would say have faith," with the "JERK" at the end of it unsaid, but, in my opinion, implied.
Captain Dameron had just gotten a bunch of people killed in a foolhardy mission for an unnecessary goal, and been demoted from squad Commander to Captain of only his own ship. Why the hell would she tell him anything, and why do you ignore that and assume she's just prejudiced for no damn reason? He even lied about his rank because his general was indisposed.
Poe earned nothing from Holdo, and should have just followed orders.
In short, the entire internal strife among the Rebels is completely and utterly Holdo's fault.
Maybe you should blame the mutineer for how badly the mutiny cocked up the Resistance's plans instead of the woman in charge.
Add on the fact that this whole movie seemed determined to undermine the acts of heroism we've had for seven movies. Luke's a hero of the Rebellion, but most of the characters (especially Rose, who couldn't seem to utter a single line without preaching in a terribly written attempt to make her seem deep, wise, and strong) would probably find his actions from the original trilogy stupid and risky.
It's almost like you understand the movie. And they would have thought that because those actions were that. The problem is in mythologizing those actions and then acting on that myth.
Luke had the Force going for him and Poe doesn't, so that's why it worked for Like and doesn't feel Poe.
Nor can the characters from the previous movie, which were bright spots—Poe and Finn, namely—have zero impact on the plot of the next one. If you removed them from this movie, literally nothing changes, except it's 45 minutes shorter.
A lot less people would have died, the fleet would have escaped earlier if the not-Poe Commander had listened to their general, and the evacuation to Krait could have succeeded. Their impact on the plot wasn't NOTHING. It was just nothing GOOD.
Hell, Finn's motivation doesn't even make sense. He wants to ditch the beacon to Rey somewhere, right? So he makes it off the ship to a planet that's decadent (where the film stops to tell you a Very Special Message about Animal Cruelty and War Profiteering is Bad Guys, Okay?) but relatively safe, and just... keeps the beacon with him. While he goes onboard the flagship of the First Order. He could've just tossed it into a trash can if that's what he wanted.
It's almost like he wanted to see Rey again. Jesus, you pick that as what doesn't make sense? It makes perfect sense! The motive is clear!