Hey Ian and Andrey,
I don't agree with your opinions you offered on unit tests, although it does sound like one of you inherited a unit test catastrophe.
Even a small, 4-screen app should have a few judicious unit tests. Not to test what happens when a button is clicked - that is not a unit test. And not to interact with a database or other systems.
The unit tests you have are, for example, to test that the code you have written that can detect which of three possible date formats is used. Or to test that any of several different dates get interpreted correctly. That is, testing a unit of code in the form of a method or function, which is what the "unit" refers to in unit testing.
Then, when a bug report occurs with a date not being handled correctly, you can add a unit test for the troublesome date, fix the problem, and be confident that your fix did not break the handling of any other dates.
This approach is beneficial in just about every product, is free of "100%-coverage" mantra, and helps rather than hinders rapid development.
My mantra is that the whole set of unit tests in your project must run in no more than a few seconds, and must not rely on you introducing major architectural changes to your project.