Is there any property in particular of dancing links that you think helps in determining this, or is it just that a backtracking search can be used to test all cases?

For pen-and-paper puzzles like Sudoku there is usually the goal that a solution should be findable by a series of deductive steps. For 9x9 Sudoku, most deductive steps used correspond to the effects well-known propagation techniques offer[1]. With a suitable propagation level, if the puzzle is solved search-free, then one knows that both there is only one solution and that there is a deductive path to solve it.

[1]: See "Sudoku as a Constraint Problem", Helmut Simonis, https://ai.dmi.unibas.ch/_files/teaching/fs21/ai/material/ai... for some data on 9x9 Sudoku difficulty and the propagation techniques that are needed for search-free solving.