KK has every right to ask for a 1,2 or 3 year deal because that's the time frame where he is under team control as per the CBA. By year 4 and going forward neither KK or the Wild have any "rights" over the other.
The only "rights" that matter are what a team or player currently has. At current, Kaprizov has very limited rights due to his own actions. His options are/were to sign his QO, sign a contract he and the Wild both agree upon, or go back to Russia until he either can agree with the Wild, or until his last three RFA years are over.
The Wild are fully within
their rights to only offer 4+ year contracts. If Kaprizov doesn't want to sign that, he is fully within his rights to either accept the QO (too late for this season) or go back to Russia for another year or three. Those are his rights.
But if he wants to play in the NHL this season and the two after, he's going to have to come to terms with the Wild. Those are the rules of the league. You may not think they're fair, you may not think the Wild should be able to do that, but it doesn't matter what you think.