Too many new pages for me to want to read. Just one point.
He is just standing there, not moving, trying to breathe as silently as possible.
Does that count as a stealth check? Nope. So no action has been taken.
Actually that is virtually the definition of stealth. You stop moving and try to be quiet (when you're in a position when you aren't clearly visible).
As annoying as we pretty much all find the rules in terms of timing and action economy, I cannot fault their definition of stealth. If you simply turn invisible and don't
actively try to be quiet, you actually probably would be noticed by a nearby person who isn't trying to notice stuff. Now, the way I interpret the rules of the game most characters do not get to even use their passive perception unless they are doing the non-combat equivalent of spending their action making Wisdom (Perception) checks each round. "Passive" is a misnomer, since you have to be actively doing it.
So, outside of combat, if you turn invisible and no one is close to you and intentionally keeping alert for out of place things, they probably won't even get a chance to notice you.
Inside of combat, it is assumed everyone has their passive perception up by default (and doesn't take an action unless they want to roll an active perception check on top of that), so they
are trying to be aware of everything going on around them, and
would notice the breathing or other signs of an invisible but not hidden creature.
Summary: The rules actually do make sense that you would still need to actively hide when invisible, but action economy makes that a pain, and One With Shadows is barely useful in combat.
Also, I don't see a good reason that the invocation would work outside of combat but not inside, since combat rounds are a meta-concept, not an in-world thing. (Sure, I can make up a lame justification, but that's not a good reason.)