It kind of depends on what you mean by there being enough information to solve it.
There's a guarantee that at least one solution exists, but not that it's unique.
There's no guarantee on exactly what kind of process you can use to find that solution.
Sometimes it leans more into logic, sometimes into pattern matching, sometimes into reasoning about what kind of n-grams appear in English, and some times playing the odds a little bit. I've played hundreds of levels of this, and not yet seen a level that felt like brute force was the only option.
This is the kind of thing I meant when saying in the description that the puzzle generation isn't as elaborate as normal. It'd be easy enough to for example guarantee unique solutions (assuming the dictionary is static), or to make guarantees about exactly how many positions the first word can be in, but that also reduces the variety, and I don't know for sure yet what kind of levels end up being the most fun.
The hints are really there to allow players to self-balance the difficulty, and to allow for an escape valve if the puzzle is/feels unfair.
