My argument is that too much freedom to interact with abilities and items completely undermines the point of puzzle design, and if a puzzle allows the player to break it any way they want then it's a classic "fake puzzle", and Nintendo doesn't want the player to get stuck or offended, so the puzzles in TOTK were designed to please the player, but as a game design they failed.
Having multiple solutions, intended or not, is not "fake".
Having an equation, wanting 1 solution, but there being multiple does not disqualify it from being an equation. Puzzle is a boat term, its not refering to a specific type of problem or game.
Perfect knowledge, inperfect knowledge?
Is there a clear defined answer, or is the solution tied to a set of rules?
Puzzled that work with social concepts need to have the answer tied and the solution can depend on social context. puzzles that work with rules either need to rely on rules in the natural world (math, physics, etc) or needs to communicate the rules properly. Figuring out what those rules are can be a puzzle on its own.
While some solutions in Zelda can be seen as "cheating" the puzzle (like solving a ring puzzle in real life by bending the rings, I would say bringing a bunch of metal weapons und Putting them all over the place could count as that), others are just a different interpretation of the rules set up to get to the solution (instead of using magnesia to move something using stasis).
Totks puzzles are designed a little to broad to be challenging. Sure. But fakes totally the wrong word.