Confirmed half right half wrong, it does appear and does fight regardless of whether its first on list or not, so you mighta missed that, but it does indeed seem to disappear after each fight. Very strange, considering its supposed to last til killed or new day. I mighta missed something too, cause I see no 'returns to wood' message anywhere, and I tried it thrice. Jukai koutan has also never changed the price of any mokuton moves for me, only the reset number has.
Either way, resets as a whole (except rinnegan and sage mode) are being worked on, and while I wasn't told the specifics I still daresay this is among the things that are getting changed for the better.
So for now just sit tight and be patient.
Edit: Was blind, jukai koutan does indeed drop the prices a little.