Everyone is so weird about apologizing to children.
Authoritarian parents, such as my least favorite person Abigail Shrier, are against apologizing to kids. You have to show that you're in control! You have to let them know who's in charge! You can't let them walk all over you! You can't admit fault or human frailties!
But I'm a hippie-dippie positive parent who believes in children's rights and being nice to kids. Hippie-dippie parents do believe in apologizing to children. But they make such a big deal about it! "On rare occasions you may need to admit fault... it will be emotionally difficult... you will need to be humble... but consider how much you're teaching your child about personal responsibility... modeling taking accountability for your actions..."
This is wild to me.
I am constantly apologizing to my child. I apologize to my child like once a day. Here is a sampling of things I have apologized to my child about recently:
I forgot to bring a towel to his swim class and we had to dry him off with paper towels.
I didn't realize the coffeeshop was closed on Mondays so we couldn't get a pastry to make up for the paper towel incident.
I ate the last slice of cake because I forgot that I'd promised that he would have it.
I accidentally hit him on the head with my laptop.
He tripped and full-body slammed into me and I was startled and yelled at him.
He was delaying putting on his shoes and I got frustrated and snapped at him.
I thought Boy Scouts started this month but actually it started next month.
I brushed his hair too hard and it hurt him.
We were out of microwaveable pad thai and oatmeal.
I am so confused about how all these parents that get this parenting advice are responding to these incidents. Are they all just very competent people who never brush hair too hard or forget to bring a towel to swim lessons? Are they saying something else when they make a mistake? What? It just seems so weird and awkward to me to respond to whacking someone in the head with a laptop with anything other than "I'm sorry, I should have looked where I was going."
Some parenting things are legitimately hard. As you can see from my list, I am doing a mediocre job at not raising my voice at my child.1 I don't know how to balance giving my child the freedom to make his own choices with protecting him from the Infinite Compulsive Slop Machine. I frame a number of things as "giving my child freedom" that a less charitable person might characterize as "laziness."
But I am mystified why it would be hard to apologize to your child. All you have to do is treat your child the same way you'd treat anyone else and say "sorry" when you've done something that annoys or inconveniences them. And yet parenting advice implies that numerous people have difficulty with this somehow. I am baffled.
It feels subconsciously like I lost control, but I definitely raise my voice more at Vasili than I do at adults who have more ability to make their anger my problem.

It really is weird. As an example, my own upbringing was generally pretty good, *except* for my father's absolute, unyielding insistence of never, ever, ever, ever admitting fault for anything, even in the direct face of obvious facts, a habit that continues to this day and completely baffles me.
You would get something very trivial, like where he'd forget to close the mosquito screen door after going outside, so a bunch of mosquitos came in the house. Then when confronted about it, there would be absurd denial that it happened, or insistence that it must have been someone else, or forced changing of the topic. Anything except admitting fault or apologizing, which would "lose face".
I can't explain that sort of reaction other than through the basic instinct to "save face" and preserve your ego overriding everything else. You generally can't do that sort of absurd reality-denying reaction to adult peers, so you don't. But you *can* do it to your children, because of the parental authority dynamics, so you do.
As more of a "mildly (ok maybe moderately) chaotically permissive" than hippy-dippy once-upon-a-time parent I'm totally with you on this. And while I think there's a point to other commenters' points of "they don't include normal "accidentally stepped on your foot" apologies" -- that the "hard" apology maybe applies to more serious moral (losing my shit and yelling) or practical (the towel incident) failures -- ie not random accidents but more "getting things obviously wrong" situations -- I think there IS more to this.
Because many of those instances you describe (and I can totally relate to) are adjacent to answering "I don't know" to a child's question. And while I always loved my children bombardment of questions, I never ever had a issue with saying "I don't know" if I didn't know the answer. Yet seemingly many, many parents do struggle with this (and tbh also many people struggle with a dk in other, non parental situations). So maybe it's something about one's confidence in one's competence?
Now, apologizing for moral failures (yelling, or similar) can be hard because they require admitting to doing a Bad Thing. Guilt etc, but still i think aren't as hard as one type of apology that maybe the stricter parents never make and the hippy dippy ones make A LOT: apologizing for doing something you don't consider wrong which nevertheless caused the child to be upset, eg dragging them home if they wanted to stay out, or subjecting them to painful or scary or socially taxing / excruciatingly boring experiences. I'm in two minds about those. I think I often did apologize but it feels kinda both pathetic AND hypocritical: I'm not really sorry for doing the thing even if I wish the child wasn't upset by it. I feel this type is the one that's most likely to lead to trad/hippie conflict too.