Is it possible to permanently remove a curse from a cursed magic item?
An example of such a cursed item would be the Berserker Axe:
You gain a +1 bonus to attack and damage rolls made with this magic weapon. In addition, while you are attuned to this weapon, your hit point maximum increases by 1 for each level you have attained.
Curse. This axe is cursed, and becoming attuned to it extends the curse to you. As long as you remain cursed, you are unwilling to part with the axe, keeping it within reach at all times. You also have disadvantage on attack rolls with weapons other than this one, unless no foe is within 60 feet of you that you can see or hear. […]
So in this case, to permanently remove the curse would presumably turn this magic item into a curse-free magic axe that just has a +1 enchantment and increases your HP whilst attuned.
The first obvious thing that springs to mind is the remove curse spell, but that explicitly states that it cannot do this:
At your touch, all curses affecting one creature or object end. If the object is a cursed magic item, its curse remains, but the spell breaks its owner's attunement to the object so it can be removed or discarded.
All it does is end the attunement so that the creature is no longer cursed, but the magic item itself is still cursed.
I will also bring attention to the special case, the Sword of Vengeance. This weapon can have its curse removed by casting the banishment spell on it, but only because it's description explicitly describes that it is possible, and how to do so:
You can break the curse in the usual ways. Alternatively, casting banishment on the sword forces the vengeful spirit to leave it. The sword then becomes a +1 weapon with no other properties.
The following Q&A explores that in more detail: How does the spell Remove Curse interact with a Sword of Vengeance?
For the purposes of this question, I want to ignore the Sword of Vengeance, since I'm asking about the generic case (or if we need a specific magic item, the Berserker Axe, which was also called out in the accepted answer to the above Q&A as a counterexample to the Sword of Vengeance).
So, is there any way that one can permanently remove the curse from a cursed magic item (besides the Sword of Vengeance)? Excluding catch-all things like wish or a cleric's Divine Intervention (unless they specifically mention removing curses from cursed magic items).
Best Answer
Within the parameters you have dictated, no methods exist within RAW
The specific rules you have already highlighted in your question, being the core rule in the remove curse spell, and the specific exception on the Sword of Vengeance. This cited question, unfortunately, chooses to cherry-pick the phrasing of SoV in a misleading way, leading one to infer that there is precedent for remove curse being able to eradicate a curse from a cursed magic item.
The general rule is that a cursed item is always cursed. An attuned creature has the item's curse extend to them, and casting remove curse on that creature gives, at best, a brief moment of relief before the curse is instantly re-applied. Casting remove curse on the cursed item breaks its attunement, ending the curse on the creature but not the item.
The only "exception" to this rule doesn't even involve removing the curse, but rather using banishment to rid a SoV of a vengeful spirit that is the source of the curse. The specific ruling on SoV overrides the general rule in that instance alone. There is still no precedent for the remove curse spell or any similar effects being able to rid a cursed item of its curse.
Unfortunately, wish and divine intervention remain as the only permanent RAW methods, as you have surmised.