Maybe you've never been in mountian lion country, so you don't know... but the fuckers are fast and deadly. They very rarely come close to humans, but this one was already in the front yard so it didn't give two shits about what would happen to it. That makes it incredibly dangerous. I'd grab my gun immediately after seeing it.
Having a family member that's unaccounted for would just make me that much more worried. Where did she go? Is she okay? I'd crack the door open and if I heard a growl I'd shoot at it.
We have the benefit of hindsight, but in the heat of the moment I don't think I can blame the guy. Having a family member unaccounted for after you just saw a cougar outside would (and SHOULD) make anybody go into "must save family mode". Deputy was quick on the trigger but IMO had the right intentions... girl was just flat out retarded.