Hence, if you somehow manage to take into account all the initially available information, then no search is required, because, in a sense, the given hints already narrow down the entire space of (ostensibly) possible assignments to a single one.
There are constraint solvers that do this. For example, with the CLP(B) version based on BDDs that I mentioned, the solver internally constructs a Binary Decision Diagram (BDD) which is like a (ideally: more compact) truth table of all assignments that satisfy the requirements.
Therefore, with the CLP(B) version I posted, search is never necessary for valid Sudoku puzzles, because they admit precisely one solution, and the constraint solver deduces this unique solution by reasoning about the decision diagram.
There's a severe drawback to this: The BDD may be exponential in the number of variables. That's a problem, because there are dozens of variables in many Sudoku puzzles. In practice, we are often lucky, and the worst-case space requirements do not arise.
For comparison, the CLP(FD/ℤ) formulation is based on constraint propagation that ensures a certain kind of consistency for all individual constraints in isolation, but not necessarily for the combination of all constraints, i.e., the problem as a whole.
In practice, at least for Sudoku puzzles, this also often leads to finding the unique solution without any search. This may also be the case if only the minimum number of clues for a valid Sudoku puzzle (i.e., 17, per https://arxiv.org/abs/1201.0749) is specified, as the parent mentions. In general, search is still required. In the concrete case of (standard) Sudoku, the required search, if there is any at all, is typically very small, because the many constraints usually narrow down the entire space considerably.
Reducing the amount of required search is the key attraction and goal of the consistency techniques that are used by constraint systems. The available systems differ in how much and how fast they are able to prune inconsistent values, leading to notions such as global consistency, domain consistency, bounds consistency etc. that give some indication of propagation strength.