If a hard fork occurs and is broadcast to the bitcoin network, is there any manual intervention for miners to accept the fork?
A hard fork means that the consensus rules of the fork are incompatible with the current consensus rules. So in order for miners to accept a hard fork, they would need to be running software that views those consensus rules as valid. This means that they have to manually intervene to change their software prior to the fork in order to accept it. So yes, there is manual intervention for a miner to accept a hard fork.