Canada

For the Ohtani clock thing you could think of it that way. When all you're doing is pitching you come out from the dugout and immediately start warming up for your half of the inning. If you're getting ready to bat you've got some gear on that takes a minute to take off. A minute lost from the normal warm up timer may be considered too much time (that may compromise safety), so they add a little bit of time to allow for that case. The same would be done for any pitcher that is also a batter or position player, but that is a fairly rare occurrence so it doesn't come up very often.

As far as the hitting of the batter is definitely suspect to throw inside three times in a row, but just on game state there's no reason to hit that batter in that situation. The only reason that warnings were even issued (in my opinion) was as a result on the bench clearing, not the hit batter in a vacuum.