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While we do fairly frequently bring up the question of what exactly qualifies something as a "chess variant," we acknowledge that there are different definitions.  For this reason I think we try to be inclusive rather than exclusive in allowed items.  (At least, that is my justification, and I suspect that may have been the case historically as well.  Also, it is helpful that this is a website and not a book: if I were writing another Encyclopedia of CV's, I would be more inclined to be exclusive for space considerations.)

Specifically here, we have a game played with a chess board and chess pieces (well, pieces with chess movement plus arrow-firing), but the goal and some mechanics have been completely changed.  (Somewhere, I can't find it now, this group of games is described.  Joust is another example, and maybe even puzzles like the 8 Queens and similar things qualify.)

I've generally thought of Crossovers differently: they are specifically a blending of Chess with some other established game.  E.g., I wouldn't consider this game or Joust crossovers.  Cheskers is a crossover with checkers, and For the Crown is a crossover with deck-building games.

Cheskers actually does seem to meet the royalty condition, though in a different way than standard chess (starting with more than one, and allowing the addition of new ones).