Unit Test could be part of the QA process but it shouldn't be the whole QA process.
In general, Unit test is used as:
a) some QA uses only Unit test and those processes usually miss a lot of troubles that can't be tested automatically. For example (and it is not so strange), lazy QA guys that ask a ready made unit test then they can input some values and call it a day.
b) Other QA uses as a compliment of other test. However, i trends (always?) to be redundant.
So unless the Unit Test is painless (specially about to not to add or modify the code) then, old-school-testing is the solution (that in any case, manual testing should be done).
