OK, an n-bit hash function where some subset can be used to provide 3..n-bit hashes, requires strong properties of "spreading the influence of the input"... otherwise the subsets won't work well.

If that is your requirement, then CRC is indeed not for you.

Testing the effectiveness of a hash is hard work, and as you say, limited by the ability to select test data. So I can see the attraction of testing some properties of the hash instead. Hopefully those properties are strongly correlated with effectiveness :-) (I suppose some confidence testing is required...)

Thanks for the references. I will now "go away" and study :-)