Hi, I am Harry Potter fanficfan. I am critical to canon aspect, see ARedHair excellent analysis of canon plotholes. My views of canon problems are below. As Harry Potter stories go you could split these to several types with certain cliches.

Good time travel are my favorite genre.These are rare to find. Best one is Delenda-Est or Passageways. In both Harry meets young Bellatrix before she went insane/mindcontrolled.

Next are time travel fics from robst who is one of most talented authors around

One of most touching stories is Backward-With-Purpose-Part-I-Always-and-Always. It almost made me like RHr ship.

Next is 30-Minutes-That-Changed-Everything where Harry meets Hermione in first 30 minutes and everything changes due new revelations.

Then a pretty good stories are Nightmares-of-Futures-Past and Harry-Potter-and-Future-s-Past

Brilliantly subverted in Harry-Potter-and-the-Freak-Parade where Luna takes care. When they go to past Harry tells Luna to alone invent mark that they cast on Dursleys and don't teach anyone else. That makes her invulnerable with Harry that is due prophecy.

Bad time travel Main problem of finding good time-travel fics is sifting among bad time travel fics. Most fall into canon-slave trap. Best seen as parody time-travel pedophile. Main problem is that author tries to preserve timeline instead let it grow naturally. And timeline is most boring path possible. Everyone already read HP books so reading about troll again gets boring quickly.

Main problem of time travel is doylist. Knowing future is incredible advantage. And it would result in pretty short story watch this.

With canon character future Harry would finish Tom in week. He would place confondus at RAF generals to make Malfoy manor and Azkaban practice targets as shields dont stop bombs yeah. Then he would told Dumbledore to collect rest horcruxes, and dose Quirrell with draught of living dead until he could kill him.

These make nice one-shots (for example The-Greatest-Minister-of-Magic) but hardly longer story.

So if you want to weaken story you need combination of upgrading opposition (best with evil Dumbledore character interpretation as good DD significantly helps Harry), limiting time-travel (Future harry could only send hour of memories/manifest as seer visions) or add artificial limitation like fate/preserving timeline (which makes poor story). Reason why third option is bad that you could make story better by omitting time travel. Instead Harry would meet mentor that does same (For example Regulus faked death, found young Harry at Dursleys to train him and told him about Horcruxes.)

Marriage law fics Mostly romance subgenre where Draco marries Hermione (or worse when Ron marries Hermione to keep Draco away) and they initially hate each other until they love each other. Not my type.
Subverted in one of best fics I read Hermione-Granger-and-the-Marriage-Law-Revolution which shows what would happen - French revolution.

It also deconstructs shields can't stop bullet and wizards fear muggles because there is too many of them and they have gun nonsense. AU where that holds would quickly evolve into what happened there. You don't need combat spells. You need army of Invisible wizards on broomstick firing miniguns. Loved ending with HHr marriage where Ginny has several hidden weapons, luckily less than Ron as he has shrunked tank.

My pet peeves when reading HP fanfiction are:

Being slave of canon When I give up story its mostly because they stay too close to canon. They consist mostly of same scenes as canon only bit modified and that gets quickly boring. Most attrocitius example are time-travel stories where characters try to preserve timeline. I am ok when canon serves as background and focus is in relationship or when its crackfic.

Instead I would like that character would act realistically and deal with consequences without railroading them back to canon.

Walls of irrelevant text Heroes newer go to toilet. I often skip paragraphs when they are of little consequence. Combined with previous point I hate to read again entire paragraphs from book. Just do quick summary to introduce a scene.
Second problem is with excessive shopping trip/trunk descriptions. Just describe plot-relevant items, I do not care about eyeliners from Gucci.

Harry Naruto Potter fics about nine tailed fox whose flick of tail could kill Voldemort. It is hard to write good powerful!harry fic as you fall to Naruto trap.
Best form of powerful Harry fics is relatively short story. Harry gets power, kills Snape and Malfoy at first chance, finds and kills Voldemort, marries Hermione, kills Dumbledore if he objects on previous point, not necessary in this order.
Sadly most fictions drag because Harry acts like idiot of Naruto's caliber. He could quickly solve problems that drag on five chapters, probably by doing one of action above.

My option on HP characters

Rowling is author that sacrificed world for drama. Her books are great story. However when you start writing fanfiction you find that how she intended character and how he actually acts are two entirely different things.
If you want JKR version watch the movies, they are IMO better than books as they omit problematic passages and characters are more likable than in books (particulary Snape, Ron and Draco).

So here is my analysis

Ron

I would say he is most conflicted character in series.
JKR: Good, loyal friend of Harry.
Books: Ron the useless. Often brings Harry into trouble. He helps him as friend should ... eh? Yes, I found it. In first year to win chess match.

He needs more moments of spotlight in books. Also most male readers, myself included are jealous because: "He stole my girlfriend." Female readers don't like him as he reminds them their idiot husband while she could be with their lovely Draco.

Ron needs some moments of awesomeness to be likable. Best story of turning him into Ron 2.0 is Fugitives of Azkaban Or other authors make Ron 2.0 by naming him Neville.

Dumbledore

His main problem is Doylist. If he could freely act while being as strong as described in book he would say on Harry's 11th birthday: "Hi Harry, you are safe now. All death eaters are in prison, Sirius is free and I just used petrification potion on Quirrell alias Voldemort." THE END. However that makes pretty short story so he cannot never help Harry to keep story going.

JKR: Kind grandfather. He is very powerful politically strong and has many connections.
Books: He just waits instead acting. When he acts he always arrives too late.

As he is used as plot device its hard to make him a consistent character. Most common explanations are:

Dark Lord Dumbledore: Sadly this is simplest explanation that fits facts. He protects DE on every chance, hires Snape to train DEjungen as he uses Voldemort as his puppet. DADA and History are terrible to make sheeple weak. Due prophecy he manipulates Harry into dangerous situations ideally being killed by Voldemort to satisfy prophecy and not being killed by Harry.

Insane/senile Dumbledore: Other variant is that he decided to fulfill prophecy, sending rest of world to hell if necessary. He hired Snape as Hogwarts needs evil potion instructor. He is unaware what happens in school as he is focused only on Prophecy and finding horcruxes. Add being religious fanatic that believes that saving a sinner is worth hundred good mans.

Albus Dolores Dumbledore: He is common in manipulative!Dumbledore fanfics. When Harry finds that he stolen his money Dumbledore makes most idiotic plan imaginable. He is surprised why Harry avoids it while shouting "I am Albus Dolores Dumbledore, Supreme Mugwump of ICW."

Manipulative Dumbledore: Most independent!Harry fanfics have him. But most of time he belongs into one of three categories above. He says that he does it for greater good. Most of time its just excuse as for true greater good he needs to select action that has best consequences. You could improve his plans in five minutes. For example he could teach Slytherins how backstab their fathers, put Malfoy in prison without trial with polyjuiced Snape shouting in diagon alley: "Dark lord will return, crucio" and so on.

Oathbound Dumbledore: There is excellent explanation of his action in Knowledge is Useful But Power is Power Dumbledore's greatest wish is start hunting death eaters and Voldemort but he can't as Chief Warlock oath forbids him to spill pure blood. He tries everything to use loopholes in his vows.

Potioned Dumbledore: Snape gives him loyalty potions, he sometimes acts correctly when its effects weaken.

Weak Dumbledore: As most plotholes were caused by Dumbledore being too strong. It could be fixed by making him weaker, Wizgamot is controlled by ex-DE. Snape and Binns teach because board of governors said so. He needs to keep basilisk secret otherwise Fudge would use it to make Umbridge headmaster, he cannot openly teach Harry for some reason. He can't tell him anything because Snape/Voldemort would use legilemency to spy on him...

Snape

JKR: Ambiguous evil (secretly brave good).
Movies: Sexy evil.
Books: Evil.

There is big difference between dream Snape and book Snape. Fanon Snape is sympathetic sexy character that is ambiguously evil as Rowling intended. In first six books he is simply evil (see below.) In last book she tries to retcon Snape as brave antihero. His problem is similar to Ron's: lack of scenes that have him in good light but there is plenty of evidence to make him evil.

Sadist incompetent teacher - why does he teach potions if he could be school healer... This is often fixed in AU where its just Harry's impression as he is bad at potions and Snape yells at him to stop blowing up cauldron.

If one always acts like death eater he is death eater - Snape openly trains and protects DEjungen, betrays Harry in PoA as he could Legimens truth, OotP where he tells Voldie that trap succeed while he has three hours to warn order but doesn't do so. In HBP he allows using vanishing cabinet instead turning it into trap. As spy he is almost useless as he doesn't help order while alive. He could play card that Dumbledore wouldn't trust him if he didn't act as reformed DE.

Harry

Nearly all fanfics are what if Harry had alternate personality. Its understandable as when one asks: What would Harry do? Canon answer would be lie on bed and be depressed.

Hermione

Is most complex character in books. Some think that she should be hero of first three books, read potato head. You can imagine parody of first three books where Hermione goes to Hogwarts with her puppets Harry and Ron and drags them everywhere.