[Celerity]

My wife points out that even if it’s a pen, carrying it at the ready seems a bit odd too.

Unless the man is trained that the current situation she is in its known to trigger. This would be cameras in her face, or something going behind the view finder

[Blue Jays]

Yes, concur with your assessment and that of your wife.
In the event Hillary Clinton needed the contents of a medical pen injected, her handler could extract it from his pocket and likely deploy it in under fifteen seconds.
She must be in a really serious situation if it needs to potentially be administered within five seconds or less.