Since not too long ago "Lack of context" is a close-reason on math.SE, yet since long ago there were guidelines how to approach and how to ask questions that are likely or explicitly homework questions. See for example this thread for a start https://math.meta.stackexchange.com/questions/11075/guideline-for-using-lack-of-context-and-homework-close-reasons
On the one hand, I mention this to counter the idea that might exist in some people's mind that math.SE is so tolerant against cheating. This is not really relevant to the debate but still I consider the description of math.SE in the question as somewhat imbalanced and thus wanted to counterbalance it a bit.
On the other hand, and more directly relevant, this means that sending questions that are only a problem and nothing else (e.g., directly copied from a HW assignment or textbook) directly to math.SE is indeed not a good idea (anymore), though for reasons somewhat other than the one in OP.
Instead one could mention math.SE with some additional explanation and guidance. This course of action is also advertized since a long time by some users active on both sites.
I cannot see anyting unethical about doing this. (Aside: even if abstractly it were, the actual effect in my opinion would anyway be rather minimal, as they likely will find the math.SE site or still another one without instructions; to give them is simply useful to get rid of the poster in a somehow friendlier or non-confrontational way.)
While I am certain some people try to use MO and math.SE to cheat and this is in some sense problematic, not each use that could be cheating in this form actually is and I consider it as not a good idea to assume automatically the worst in each case (even if this should be the typical case, which might or might no be the case).
More generally, the idea that "we" are responsible to prevent cheating on HW, in my opinion, is fundamentally flawed; not only but also since we cannot effectively prevent it, and in trying too hard more harm than good is done.