I've always heard that "security through obscurity" is discouraged because, well, there's no stopping someone from digging in and figuring it out.

This should tell you enough about the person.

Obscurity very often increases security. The question is - by how much?

It is fine to add obscurity AS ANOTHER layer.

It is the same story as the "open source software is safer"

No. If you are open source & you have significant community then it is, otherwise closed source is harder to attack.