I agree that such exercises have a built in risk of giving the students the wrong idea what is the generic situation and what is quite a special case.
However, "the risk that the student knows she is wrong when some crooked formula/value shows up" you mention is more a feature in my opinion. It is not a bad thing to do heuristic checks that suggest one might have made an error and to then double-check.
Also, for some things it is more or less unavoidable to present constructed exercises. A polynomial will not have a multiple root just so, you need to take care it has one.
Or, in linear algebra one has to choose the matrices somewhat carefully if one wants to illustrate certain things. And, it is a real pain to calculate even with moderately sized matrices if they involve radical expressions, and if you go for decimal approximations you 'lose' the clear notion of singularity of a matrix and would actually need to do something more advanced. So, somehow one is stuck with matrices whose eigenvalues are "too nice" (and this even leaving the issue of factoring the polynomial aside.)
Personally, I mostly tend to give exercises where things work out nicely, yet I do talk with students about the fact that this is what happens and that this does not capture what would be a "typical" situation were one to choose things "randomly."
That is, I think it is important to raise awareness of the issue of "artificially nice" problems with students, but as mentioned above consider it as unavoidable to use them in certain cases.