I think it is reasonable to take everyday speech saying "does not exist" as meaning "appears not to exist, and if it does exist this is indistinguishable from not existing".
Come to think if it, I'm pretty sure I've never stated "invisible pink unicorns do not exist". On the other hand, I've never had to avoid saying it either; oddly enough, I've never felt any urge to comment on the possible existence of invisible pink unicorns until now.
But this is immaterial to the issue at hand, which is one of logical arguments, not mere validity of statements. I don't believe that invisible pink unicorns exist, but I would never state that I had logically proven that invisible pink unicorns do not exist.
The IPU is a poor analogy for disproving the existence of God because it gives up too much ground.
God is in the same category as a square circle. Something that is self-contradictory and therefore cannot exist.