I really don't want my children to apologize if they don't mean it and I really want them to apologize and mean it.
So I think it's about modelling(it's always about modelling) what you want to see...teaching, explaining, identifying the feelings and deciding what you want your child to do.
Generally while my kids are small I will identify their emotions for them...you were angry so you hit so and so...we don't hit, it makes so and so hurt...then after that I apologize...I would say "I am sorry you got hurt"
I have been on the recieving end of apologies that meant absolutely nothing or were used to get the person out of scrapes...my ex husband for example...he was always sorry...for getting caught but not for the bother he caused others. He apologized daily pretty much and I stopped accepting his apologies...accepting them meant he got away with it...and he would do it again.
I want my kids to be able to deal with real emotions and real relationships in a healthy way and while I want them to be polite I really want their personal relationships to be authentic..that's the most important thing...
I have had a harder time with my older kids..who were forced to apologize for everything by their dad..they can be extremely polite but never apologize when they wrong someone because they don't get that sorry means you "feel" sorry it's not just a word.
They are learning now though because I have talked to them about it...at 12 and 16 I have to really work hard to get them to not try to wiggle out of situations with an automatic sorry(schools force it too). My oldest had a fight with a girl and they had to apologize to each other...school dropped the issue, girls hate each other still and nothing was resolved because sorry meant nothing to either of them. My dd is not the least bit sorry...and no she's not a sociopath, she had been tortured by this girl for a year and felt she deserved it, still does. I am sure eventually she will let this go but until then she's not sorry and neither is the other girl.
It's a balancing act but I have decided I won't force sorry...I will talk to them, identify the feelings, model sorry to them..ie...I apologise if I hurt them etc....but I will not force it.
My three year old is definitely getting it...she will immediately hug her baby sister if she knocks her down by accident..that genuine sorry means so much and is well worth the risk that she will be seen as impolite by random strangers.
I am sure as we venture out into the world and she sees me use "sorry" as a simple meaningless platitude that "society" sometimes demands of us she will learn to use it. In the meantime I want her to be genuine and feel sorry.
As an added note, I don't think there is anything wrong with reminding your child to say sorry as long as that's not all you do...if you teach them, explain why, talk about the feelings.
I know a lady who immediately demands her kids apologise...they say sorry as an automatic response to everything...they bump into the train table and apologise to it