Battle Step starts with an attack declaration.
Well, I guess that would still fit in, as the Draw Phase similarly begins with a draw.
But if both players pass on the chain and no effects are trying to resolve, what specifically has occurred at the conclusion of Damage Sub-step 6 that prevents activation of
Michizure? Is it the actual step change that causes you to miss the timing? Is, in this instance, exiting the Damage Step and reentering the Battle Step in actuality the last thing to happen and not the sending of a monster to the Graveyard?
The ending of a phase (or in this case, a step) is actually when there is no Last Event and both players pass normal turn priority. When the monster is sent to the graveyard, both players get a chance to respond to that as the Last Event.
If either player chooses to respond to the event, or a triggered effect responds to the event, then that Response chain would have to resolve and it would create a new Last Event. Then both players would get a chance to respond, and if either one chooses to or if there is a triggered effect, a new chain and new Last Event would be created, until there is no more triggered effect and neither player wants to respond to a Last Event.
If both players pass on responding to a Last Event, then that Last Event is cleared from the game's memory. Then, the turn player would be allowed to activate an effect free from a Response chain (this is the only time you would be able to play a Speed 1 Magic Card or play a monster from hand, or change a monster's mode, by the way).
If the turn player declines to activate an effect and the opponent declines to activate an effect, then the turn (or phase) would end. There would be no triggered effects to worry about, as there is no Last Event, and thus, no chain, that has triggered them.