Hello, Julia here.
Imagine eating a bowl of grapes. Delicious, seedless green grapes with skin so tight they're practically exploding in your mouth. Then BAM you get a mushy one and even though you spit it out immediately you still want to scrub the entire experience from your mouth.
So yes, consistency matters.
HOWEVER. Consistency does not mean static! Let me show you what I mean by comparing Steve Rogers (Marvel) with Zuko (Avatar). While, Steve across the MCU is a debatably static, yet inconsistent character, Zuko is consistent while still undergoing a dramatic character arc across three seasons. Also, beware of spoilers ahead for any Captain America movie, Avengers movie, or season of Avatar: The Last Airbender.
Steve Rogers/Captain America (MCU)
Steve starts out as a shrimpy kid from Brooklyn who is always sticking up for the little guy and getting into fights his best friend Bucky usually has to get him out of. After Steve takes the super soldier serum, he becomes Captain America and continues to be a chaotic force of good, saying screw the law, I'm going to do what I believe is right, i.e. breaking into a Nazi stronghold to bust out his friends despite it being so far behind enemy lines it's basically a suicide mission (spoiler alert, he lives. Good ol' plot armor). Steve makes a heroic sacrifice which leads to him being frozen in ice for decades, never aging, while almost everyone else he knows grows old and dies.
HOWEVER this is when things get screwy. After Infinity War/Endgame Steve is given the assignment to return the Infinity Stones back to their original place in the timeline, one of which is a time in which his long-lost love Peggy is still alive. And so DESPITE promising Bucky, his homeboy, he would be right back, Steve decides to ditch his PTSD'd best friend in the future (where they are literally the only ones who know what it's like to be a hundred years old in a modern world where all your other friends have died) so that he can go back to the past and marry Peggy and grow old together.
Did I mention that he kissed Peggy's niece LIKE AN HOUR AFTER PEGGY'S FUNERAL?? Also, there's a whole show about Peggy moving on after Steve's "death" where she slowly falls in love with this other guy (who, by the way, is an absolute sweetheart). It's canonically stated in the Captain America movies that Peggy moved on and married someone else and had a family. So why does Steve, Mr Bros-Before-Hoes, go back in time and marry Peggy? BECAUSE IT'S INCONSISTENT WRITING. It really takes his character arc, which was all about moving on and adjusting to the future, and flushes it down the toilet.
ANYWAYS so yeah, due to his unchanging morals and underplanned heroism, Steve's character arc is pretty static (definitely not as dramatic a change as Iron Man or Zuko). However, due to the writers "fan service" (aka throwing darts at a list of plot points), that character arc reached an unsatisfying ending that made viewers everywhere go "what? He would never do that!!"
Zuko (ATLA)
Zuko starts out in season 1 as a dramatic, whiny teenager with a bad haircut whose only goal in life is to capture the Avatar to restore his honor. Over the course of the show, he learns that the people he's trying so hard to please have no honor themselves and are not worthy of his loyalty. After facing his father Fire Lord Ozai (who purposely burned half his sons face off to teach him a lesson when he was like, twelve), Zuko teams up with Avatar Aang to help him master all four elements. Thanks to Zuko, Aang defeats the fire lord, brings balance to the world, and Zuko becomes the next fire lord, a mature, compassionate, and honorable leader.
It's a pretty wild ride, and the road to redemption is not without its ups and downs, but Zuko has what is considered one of the best examples of a reception arc that doesn't end in death (as opposed to, say, Darth Vader or Kylo Ren. Disney has a problem).
While Zuko undergoes this dramatic transformation, he remains a consistent character throughout. His desire for honor is a strong motivating force that remains unchanged, even though his definition of honor changes throughout the show. Even when he fails to do the right thing and chooses his manipulative sister Azula over his mentor uncle Iroh at the end of season 2, it's still motivated by a desire for honor because he believes doing so will put him back in his father's good graces. His choices never feel forced or unnatural- everything he does is what his character would do.
THAT'S what makes a consistent (and loved) character.
'til the end of the line,
Julia