I did not need to do it with grown children, but I did need to do it with some for whom I was an "authority figure". These were some of the most beneficial exchanges of all. Not as much for me but for them. None of them were offended. To the contrary. The offer itself was even more greatly appreciated, I think at least in part because they knew it was difficult for me to make.
I found it best to tell people that I did not think a simple apology was sufficient to help repair the damage, but that I wanted to DO something to help repair the damage I had done. I was surprised how often I was told that the apology itself was sufficient. Often the requests were largely symbolic.
Perhaps try out this process with someone outside the family to see how it goes. The most essential part of the process is to communicate your sincerity when making the offer. If you do, the world will indeed be a better place.