IMO, the X-Men movies are more emotionally grounded because the X-men stories are that way - and the movies that work, work because they hired some great actors in the key roles (e.g., the Xaviers, Magnetos, and of course Jackman). And those characters are fairly true to their origins. The rest - not so much.
Examples just off the top of my head: Mystique is pretty much an out and out villain in the comics; the new movies make her into a hero and leader of the X-Men. Cyclops in the comics is a tactical master-mind and Leader who can defeat all but the most powerful mutants 1-on-1 simply because he is smart enough to exploit their weaknesses and his own strengths - he’s basically the team’s Captain America but with actually useful powers of his own. He’s also the first X-Man, much like Cap is the first Avenger. None of that is evident in any of the film’s, where he essentially exists to mope over Jean and die. Iceman has always been the most immature X-Man in the comics. In the movies, he’s the most adult. Neither of the movie Angels have any connection with their comic counterpart other than the mutation. Colossus is just a non-entity (unless you consider Deadpool as party of continuity, which brings it’s own set of issues…). Though probably the most screwed-over character is Kitty Pryde, who has one of her signature comics book arcs - “Days of Future Past” - turned into yet another Wolverine movie.
That’s what I mean about them not caring about the characters, like at all. They exist to be slotted into whatever role suits the story they’re telling for that specific movie, and then to be discarded. The MCU doesn’t introduce superheroes unless they have something to do in the story. The X-Men universe is chock-full of pointless mutants. At heart, the early movies are basically star vehicles for Jackman. The latter try to diversify a bit, though in practice they’ve been pretty heavily dominated by Jenifer Lawrence (thus the complete change in the Mystique character to accomodate her star power).
As for the incoherence of the stories, the standout one’s that come to mind are Xavier’s disappearing/reappearing paralysis, same with pretty much anything related to Wolverine if you think about it, Beast’s surprise at the mutant cure in X3 (when the new movies show him essentially having developed one himself at a young age), the question of who built Cerebro (Magneto in X-Men, Beast in First Class), and of course Bolivar Trask (the imposing Bill Duke in X3… to be replaced by Peter Dinkage in Days of Future Past). Also, the age of most characters is just nonsene. E.g., Cyclops appears as a teenager in the Wolverine (set in 1970s), yet is a 20-30 year old in X-Men (set in the 2000s). Professor X doesn’t age noticeable from First Class (1962) to Apocalypse (1980s). Magneto is a teenager in 1943 which would make Fassbender’s Magneto >50 yrs old in Apocalypse.
Of course, the easy answer to that is just to say “Eh, who cares? They’re just movies”, which is a perfectly reasonable reaction, but it also makes it hard to become particularly invested in the movies as a series.
Logan is great as a stand-alone movie, btw, but it also postulates a future where all Mutants have been wiped out. Which kind of makes the events of Dark Phoenix pointless if you think about it, because no matter what happens in that film, they’re all dead by the time of Logan.