Pit Automaton has received an update to its Oracle text. Specifically, the delayed triggered ability doesn’t trigger if you activate an exhaust ability that is also a mana ability.

If an ability triggers whenever you activate an exhaust ability, that ability resolves before the exhaust ability resolves.

If an exhaust ability of a permanent is activated, and then that permanent leaves the battlefield and returns to the battlefield, it becomes a new object so its exhaust ability can be activated again.

Exhaust abilities can be activated any time you could normally activate an ability.