In general, when you ready hero, do all his attachments also ready? I think no, but I just want to check.
No, a hero and their enhancement's readiness status is completely independent from each other - a good thing too, if tapping an enhancement tapped the hero (the other side of the coin re: your question, basically), things like Steward of Gondor would be a lot less good and things like Unexpected Courage wouldn't even be sensical.
Also, as far as I can tell offhand, there is not even an effect in the game anywhere that would allow you to untap an enhancement, only ones affecting characters/heroes/allies seem to exist. I'm sure Elrond would like the introduction of such an effect very much.
Edit: Mind you, Unexpected Courage on Elrond is still great, allowing him to actually use his considerable stats instead of just being a ring engine.