A very long road
Roy points out Will Shipleys post, Will I have a few things to say.
Unit testing will help you write better software, in Agile TDD you have one of the powerful development tools are your disposal. Testing does not stop and start at unit testing, it takes on board functional, system and UAT testing as well. To suggest that only you can test a software product and do so by resorting to bashing the keyboard suggests someone that does not understand that such a test cannot be repeated (if such a process was to find a bug, how could you fix that bug and know 100% that your orginal test will check that your fix has worked). It also indicates that with no unit tests how can code be refactored or altered with out the fear of introducing new bugs or breaking old code.
To document your test cases to repeat your tests means manually running through each one for every change you make, what happens if I have 500 test cases? With a unit test suite I could test one level and with automated functional testing the next level and it can do in it seconds, leaving me free to fix the next bug.
Beta testing is a great way of testing, but it should be the last level of testing not the first. Sure enough humans can find bugs and can test things in ways your test suite cannot but your assuming that bugs will be logged and that humans can test every single little thing. With a test suite your testing every single nook and crany of your code, shipping a stable product that your beta testers (and lets face it potential customers) can use with out it crashing. They in turn can report any finge issues that you can feed back into your test suite (thats the aim so you can fix and test and not send it back to the customer broke again). Any thing less than this and your going to give your customer a bad picture of your product, any rational response to this should be
I CARE