Saying that a search "must be related to some legitimate goal of government" is no limitation at all. I would agree that searches incident to arrest (for a cause other than refusing a voluntary search) are not required (by the constitution) to have a warrant, but am not sure what other un-warranted searches are reasonable. The kings of England would certainly say that preventing assassinations and insurrections was "some legitimate goal of government", but it seems the founders disagreed that kings should be ransacking the houses of their subjects.

Yes, I agree that the states were not delegated all powers not expressly delegated by the constitution, but "the people and the states" were. I have never heard a better explanation for the 10th amendment than to say that it was restating the enumerated powers doctrine (though I cannot recall where I first heard that explanation). If you have a better explanation of the 10th amendment, I am very interested to learn what it is.