Interestingly, we see an okudagram in PRO that confirms prime Harry hit Lieutenant by 2384. Unfortunately, I can’t post it, as my instance is updating my image storage infrastructure right now.
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They threw him a bone for the upcoming Academy.
Downgraded him to Admin Assistant Harry Kim.

"Ensign Kim, we hope you are readjusting well to life on Earth. We have been reviewing the reports and are finding you quite the accomplished Ensign, you were instrumental to the survival of voyager and her crew, even taking on captain duties at times. You were basically 6th or 7th in command right after the doctor. We feel continuing to call you Ensign Kim would be a disservice to all you have achieved in the past 7 years aboard Voyager.
Congratulations Super-Ensign Kim. You are dismissed."
Thats Harry Fuckin Kim to you!
Fair. Mi scusi.

All jokes aside, was there a logical reason Kim was never promoted? Either in-universe or from a production standpoint? He definitely seemed to have a disproportionate amount of responsibility.
I could be mistaken but in the beginning of the show didn't Janeway say that promotions are indefinitely suspended? They don't know how long they'll be out there, and unfortunately there will always be ensign work
They made up an excuse about how there was 'no room' for promotions, except they demoted Paris for a while which would have freed up a spot for Kim. Also, Kim being an ensign when he's in charge of all of Ops doesn't make any sense; he has lieutenants and possibly lieutenant commanders under him.
Out of universe, Berman said that someone absolutely had to be the rookie, even after seven years in deep space, so Kim couldn't get promoted or meaningfully grow. He also tried to fire Wang a few times for nebulous reasons, only backing off when Wang got a 'sexiest man on TV' award. Wang was also the only Star Trek cast member to ever be denied the opportunity to direct an episode, courtesy of Berman. Wang also once said of Berman, "if there is a hell in this existence that we have that people can go to, he's first on the list."
So, short answer is that Berman is scum.
Tuvok was promoted. So was Paris, granted that was after having been demoted.
"We're suspending all promotions...except for my close friends of course"
He sat in the chair on "night watch," broke the Prime Directive even.
My thought is that him being "Phase shift Harry" from mid season 2 onward, he had to put in the requisite time like a new cadet. He already has the training/knowledge and kept the rank he was at. However, regulations being what they are, he's stuck at ensign for a while longer.
Broke the Prime Directive and didn't end up as Captain? Wow, things certainly have changed / will change or whatever depending on when Discovery wants to set itself. Didn't Burnham do that (or start a war) and make Captain like 10 minutes later for some hand-wavy reason?
Standards have fallen.
Regarding the USS Discovery, started out a prisoner (life sentance) for comitting mutiny and worked her way up to captain somehow. Time travel and a redacted personnel file played a part. Eventually made admiral.

They done Harry dirty
Even when Lower Decks brought it to the forefront, they showed that a promoted Kim is an evil Kim.
Actually, it could be a “great filter” situation; mostly only ensign Kims end up in situations where they end up accidentally switching multiverses. Lieutenant Kims are far less likely to end up in such a situation, thus the small sample size.
Lies! Lies and cartoonery!
I think you're drawing some conclusions there, pal
Know what im watching tonight