AITK for not greeting my old teachers and just walking past them when I went to pick up my brother?

So yeah. I (20F) went to pick up my younger brother from our old school. I studied there too, passed out about four years ago. His exams had just gotten over, and I got really late that day because I was stuck with something. I was already anxious thinking he might be sitting all alone or feeling left out.

When I reached, I saw a few of my old teachers standing around. For a second I thought I should say hi or greet them or something. But I was already late and kinda panicking, so I thought I’d pick up my brother first and if I saw them again, I’d greet them then.

Now here’s what happened. As I walked past them, one of the teachers, someone who’s always been a bit of a bully and has a huge superiority complex, started talking loudly about me. Saying I was disrespectful, I studied there but didn’t learn any manners, stuff like that. I could hear it clearly and she was saying it to the other teachers like she wanted me to hear it.

I didn’t react, just picked up my brother and left.

Later at home, I actually felt pretty bad. It kind of felt like being back in school again, getting randomly targeted for no reason. I told some of my friends and they sided with me completely. They said I didn’t owe those teachers anything and that the comments were unnecessary and rude.

But when I told a couple of family members, they were like I was being rude and cold. That I should have greeted them at least, or said sorry for being late or something. One even said I was being cruel and disrespectful, and made it sound like I was in the wrong.

Now I’m just confused. I didn’t mean to be rude, I was just in a hurry and worried. And after hearing what that teacher said, I really didn’t feel like being fake polite.

So yeah. AITK for not greeting them and walking past?