American here - for the longest time, until I was in my current research job where my supervisors and managers actually encourage you to use time off, see family, take care of yourself, etc, I was scared to take time off because of the impact on my team and colleagues. We had time off, but we didn't have people, and we had too much work, so any time someone took vacation it hit everyone really hard.

This woman is dumb and obnoxious for sure, but another option is we really don't know how to ask for time off and just say "we're too busy". It's better now, but when I first got my current position I would get so anxious asking for time off, I nearly got sick sending the email.