Well, for a program of sufficient complexity there'll probably always be some aspect that can only be checked easily by unit testing--if only stuff where the program works as intended but the programmer's intent was mistaken.
Could you give an example of that? I'm not sure what you mean.
If the programmer's intent was mistaken, then the program needs to be re-written. You don't really need unit-tests for that, you just need to think about it. Maybe the unit-tests force you to think while writing them?
"Force you to think" is basically the idea, yes. The goal is to catch places where your high-level conceptual understanding of the task to be accomplished doesn't match the lower-level model of "how to get from here to there" you've mentally constructed, usually at weird corner cases.
There's roughly three places where a programmer can make a mistake--understanding a task to be done, creating an algorithm to perform the task, and describing the algorithm to the computer. Type checks help insure that the algorithm you intended to code is what you actually coded; tests can help ensure that the algorithm you coded actually performs the task you think it does.
"Beware of bugs in the above code; I have only proved it correct, not tried it." -- D. Knuth
