Real engineers are like true Scotsmen, because you can never have enough process.

In this case, of course tests really do prove the absence of bugs rather than just (not) proving their presence.

I really don't like the whole "unit tests lets you not worry about breaking things" view. That's not confidence, it's overconfidence. And it doesn't even mostly work for things where "correct" is more fuzzy than "eventually produces this exact output".