> Do not fire unless fired upon is common rule of engagement for military operating under much more adverse conditions.

No, there are different ROEs depending on the situation. "Don't fire unless fired upon" applies to a checkpoint or other defensive situation.

If you're doing a cordon and search, which is very roughly equivalent to what SWAT does, you shoot if you see weapons.

The process to clear a room is a team (typically of four) stacks against the door, breaches the door, and then the goal is to get into the room and eliminate targets in your assigned sector.

You absolutely would fire preemptively because when you're going into a room there's just no time to wait, and you're relying on speed and ferocity to overcome the fact that you just dived into a room.

Now, all that said, that's also why you want to do that sort of thing as little as possible.

Source: US Army doctrine and training as combat arms. Thankfully never had to do one myself...