Earlier quoted context omitted.
I have a somewhat different experience. I've found TDD is useful on even tiny projects because using TDD forces you to write better code. It's really hard to reliably test things that use "magic" and side effects that affect parts of your code they shouldn't really be affecting, and really easy to test things that have well-defined and properly documented interfaces that only do one thing. Consequently writing tests,…
I would humbly suggest that because you and parent (and other people here) have such a different positive experience with TDD, that you cannot quite agree when it works, only that it sometimes works, then this is a big indication that TDD is just a placebo. But maybe that's what's needed - a placebo to feel you better as a programmer, and thus making you more productive through your feelings.
This is my experience. I've watched teams do TDD, only to come up against their first refactor and have to change 80% or more of the tests. Just the other day, they were griping about a code review that they had to do which involved changes to 2,000 lines of test code over a code change with minimal functionality change.
TDD has the effect of making you think differently about how you code. Thinking about how you write code is a good thing, but you don't need TDD to make you think differently about your code. You just need to think.