There is nothing wrong with saying "It is not open source but you can freely use the binaries". That is the same thing, but upfront about it, but this feels like open-source-washing.

Nothing wrong with that at all. But that's not what was said. Answering the question (that you posed to yourself in your own FAQ): "Is it Open Source?" by stating: "we've licensed the binary using an Open Source license" seems a bit disingenuous. A more matter-of-factly answer would have been: "No".