First off, I would love it if people took unit testing more seriously and automated it! In my view that helps greatly for getting robust software. However, not for all tests automating is the answer. Especially when you get to acceptance testing (where we validate, rather than verify) or when you do integration testing for systems that communicate with other systems it is not a silver bullet. Aside from feasibility, as automating is time consuming, there are more drawbacks. Automating all tests means assuming that you can anticipate everything that can be wrong and even anticipate every test that should be done just based on the specs. Exactly the context driven testers (such as James Bach and Michael Bolton) believe you can't, we humans are not 'wired' that way. In fact they even are of the opinion (as am I) that the best testing is done by people that design tests to a great deal as you go while testing as long as these are skilled people that understand how the software should work.
Rigorously testing software via automated tests, please do. But in my view there should always still be people that test and see for themselves. The automated tests can and will miss things that are plain obvious to human testers.