Should you apologize to children?

I once had a friend who thought you should never apologize to your kids. She thought it made you look weak, and would undermine your authority.
I argued how else would you teach your child to apologize when they’re wrong?

You know I love my grandkids a little, but over the weekend I messed up.
When my 8 year old granddaughter was 3 she got her head stuck in a stair railing. Dad got video of it.
It is hilariously poignant.
Every time I ask her if she wants to see it she is adamant that she does not.
Long story short, I came across a clip of a puppy who seemed to have his head stuck between the bars of his cage. He didn’t, but he thought he did.
I sent it to my DIL and said “Show this to Zoey!”
Rick happened to be there at the moment and apparently Zoey almost burst into tears and cried out “Mean, Gramma!” and ran out of the room.
O. M. G. I had no idea it would upset her like that.
I started scrambling to find a way to apologize.
I sent Rick to Walmart for flowers and a card. I wrote how sorry I was, and Rick ran the card and flowers over to her.
She smiled, so that’s a good sign.

I am very curious though. Does she remember it? Was she terrified? Embarrassed?