If my child broke something inexpensive and everyday sort of stuff I would certainly apologise (and ask the child to) and offer to replace it.

If they broke something dear to you, I would be absolutely mortified and don't imagine I could apologise enough but TBH depending on what it was I'm not sure an offer to replace would be appropriate. I have a few items that are either very expensive or worth next to nothing in monetary terms, but the sentimental value of them is priceless. If it were just an expensive thing, I would offer to replace it. If it were something sentimental I think I'd just be apologising for the rest of my life.

I would be very disappointed to not have an apology at all. Accidents do happen, but that doesn't mean you just pretend it didn't happen.