Ok, so Frozen is a good movie, it's a fun movie, it has themes that need to be expressed (suppressing who you are and staying in the closet your whole life isn't a good idea, don't jump into a lifetime commitment on first impressions, sometimes bad people are obvious but/and sometimes they are not, non-romantic love matters too, so on) and if you don't go in with inflated expectations you'll probably find that the good outweighs the bad.
But the entire plot hinges on the troll who knows his magic making a hideously bad suggestion (in his defense, based on the troll's later musical number I don't think trolls understand humans all that well) and the king and queen going with it thus forever changing their two daughters lives and leaving them both emotionally screwed up.
One is terrified of herself and worried that any human contact could cause her to accidentally hurt someone.
The other is left so isolated that she can be easily taken advantage of very easily by the first person who manages to get close because she too lacks human contact and the only conversation she has is with inanimate objects.
This is not, in any way, a good decision.
A better one might be:
Wise old troll: I recommend removing all magic, even memories of magic.
Queen: Thank you for the recommendation, but please only do what is necessary to heal Anna, nothing more.
Queen: If Anna can't remember Elsa's magic then she won't see the danger and won't understand why we do the things we'll have to do. It would be like begging her to be hurt again. If we want to keep her safe it's best if she remembers exactly what happened.
King: That's a good point dear, I hadn't thought of that. (Aside:) I wonder why such an obvious point escaped me.
Queen: *cough*because plot*cough*
No one really heard the story of how Anna got hurt because Anna couldn't remember, the parents just assumed that Elsa had done bad things, and Elsa felt guilty. Anna may be young when it happens, but she might be old enough to say, "It was my fault," and, "I'll be more careful next time," and, you know, "Sorry."
One can then imagine Elsa having a very different upbringing, one where, "Conceal don't feel," had a counterpoint of a strong relationship with Anna in which Anna encouraged Elsa to try to control her powers by learning to use them. It would be nice to see where the queen fit into everything. She does have a listed voice actor on IMDb, so she definitely spoke. One line as I recall. Maybe two but I'm betting less than ten words total. Not really enough. (Looked it up, less than five words.)
But, anyway, this would result in a different Elsa and a different Anna. Anna might still be lonely and looking for true love, but not because of total isolation. She'd still live in a largely empty palace because the idea was for the gates to reopen when Elsa gained control of her power which, for the movie to happen, she won't. But if Anna and Elsa actually talked to each other Elsa would probably arrange for Anna to get outside so she wouldn't be so down.
For that matter one can imagine Elsa getting out too. Isolated places would be good places for her to practice.
While the major ship in the fandom may be incest it has been pointed out that Kristoff's life is ice ("it's a palace made of ice. Ice is my life,") and Elsa has ice powers and yet the two barely meet. I honestly don't care about the shipping but later bullshit I'm going to introduce could use the idea of there being an established maybe-true-love for Elsa.
So as Elsa, Anna, and Kristoff grow up Anna is helping Elsa with her power, the key is that Elsa doesn't realize that love is the way to control it. She assumes that better control with Anna giving her a hug is because there's more comfort and less fear. Elsa meets Kristoff and starts to develop a relationship. Anna dates but doesn't ever find the one.
I picture this happening at some point:
Elsa: You don't think I'm a freak?
Kristoff: All of my family members are trolls.
Kristoff: I was adopted.
Hans takes a much more active role in screwing things over because the less closed off sisters would be more susceptible to foreign intelligence gathering so while commoners might not know a prince could find out that Elsa had ice powers and Anna was searching for true love.
Instead of wanting to get married Hans and Anna just want permission to date/court/whatever the term is publicly (needed because political ramifications could be huge) which Elsa would grant and in the process get close enough to Elsa to deliver some kind of poison/curse that he got from a woodcarver who could do subjects other than bears.
Shortly after Elsa loses control and runs away just like in the normal movie.
Since Anna isn't so easy to manipulate in this version I'm thinking she wouldn't leave Hans in charge, she'd just expect him to help whomever she did leave in charge. Which, since she's in the movie anyway (Disney loves making reference to itself) I'm thinking should be Rapunzel. She's got a reputation as a good ruler, a kingdom of her own so she won't try to take over, and so forth.
Anna goes after Elsa, probably more or less as it is in the movie, and a varriation on the same accidental harming of Anna but with some different details like:
--Anna makes a point of the fact that Elsa's control was getting stronger and stronger before everything went wrong
--Elsa saying that for some reason she's lost control and it's only getting worse and trying very hard not to face Anna for fear of ice shards. In the end it doesn't matter and Anna gets hit.
--Kristoff and Elsa are surprised to meet each other there and he says something like, "She's your sister? You're that Elsa?" having not known that he was dating the queen.
--Elsa, "Were you telling the truth about being raised by trolls? ... Get my sister to them. Now! They know how to save her."
Just like in the movie Elsa ends up captured by Hans, unlike in the movie the trolls do not break out into a totally inappropriate song and dance.
Since they know Kristoff is with Elsa, when they find out she isn't Elsa, Anna and Kristoff have time to explain some things. The end explanation works out like this:
--Anna needs an act of true love to unfreeze her heart.
--Elsa is losing control because poison/curse has turned her power against her and ice shards will eventually overpower her heart which will put her in the same position as Anna.
--Anna will die first without help.
--Kristoff takes Anna back to Arendelle because Hans seems to be her best bet, with the intention of turning around and heading back to Elsa as soon as she's dropped off.
Note that I didn't say the assumption of a true love's kiss being the thing to save the afflicted had been changed.
When Anna gets dropped off more or less the same thing happens except Hans in his, "I've been pretending to be good so much I finally need to allow myself an evil monologue," speech confesses to poisoning/cursing Elsa so that he could appear the hero, use Anna's post Elsa-implosion emotional brokenness to steer her into a legally binding relationship, and thus take over.
When Hans reports that Anna died Rapunzel, still in charge, wants to see the body, is asking questions, and isn't ready to put Hans in charge on the basis of him saying Anna said her wedding vows, with no witnesses, before dying.
Hans is essentially able to pull a coup by getting those in charge of implementing orders to listen to him over Rapunzel and thus "with a heavy heart" order Elsa's execution.
Rapunzel goes to Flynn* (her boyfriend/husband depending on timing of the movies relative to each other) who is initially goofy until he sees that she's brought him a frying pan, then he's instantly serious.
Rapunzel sends Flynn to free Elsa and warn her of the plan to execute her, and sets off to find out what happened to Anna herself.
When she finds Anna, Anna has given up hope for herself and is more concerned with Elsa because she's still got "True love's kiss" on her mind, assumes Kristoff is Elsa's true love, and know he's headed in the wrong direction (back to the North Mountain when Elsa is in the castle.)
Elsa won't let Flynn help her because she's afraid she'll hurt him but breaks out on her own.
Kristoff sees the storm and turns back, with Rapunzel's help the dying Anna tries to reach Kristoff so he'll know more than the general direction of Elsa. (And Elsa doesn't even know she needs an act of true love.)
Being told Anna is dead causes Elsa to fall into utter despair (same as the movie) which greatly speeds the freezing of her own heart (new thing.)
When Anna runs to save Elsa at the cost of her own life, Elsa notices and tries to stop Anna though it means her death (symetry) at which point they both freeze, magic boom knocks out Hans (like in the movie) and then the two thaw because each of them trying to sacrifice herself for the other is the act of true love needed to thaw their respective hearts.
Elsa finally realizes that love is the key to controlling her power.
Everyone makes their way to a boat so they won't fall in the water. Flynn may or may not show up with a frying pan.
The fjord is melted, happy ending for all.
Romantic relationships are left open. Kristoff and Elsa still don't know if they're really right for each other, Anna is still single and looking for the right person. As with the actual movie this isn't about sticking a wedding at the end.
Rapunzel and Flynn head back home just like everyone else who came from away for the coronation.
* I know, I know, his name is really Eugene. I think of him as Flynn.