Unit testing is about testing discrete units of code – a single method, no more.
The moment you go into CRUD, you are talking about testing network, IO, database and other things – this is beyond what unit testing is about. In this case, this is called integration testing (you are testing how your code integrates with other code/systems).
There is place for both types of testing (and other types – regression, performance etc…) in any software project, CRUD application or not.