A year later, I now think I have an answer to this question: No! In particular, unit tests will always be necessary and useful for regression tests, in which a test is attached to a bug report and lives on in the codebase to prevent that bug from ever coming back.
However, I suspect that any unit test can be replaced with a test whose inputs are randomly generated. Even in the case of imperative code, the “input” is the order of imperative statements you need to make. Of course, whether or not it’s worth creating the random data generator, and whether or not you can make the random data generator have the right distribution is another question. Unit testing is simply a degenerate case where the random generator always gives the same result.