My 3 year old son is quick to anger. He's mellowed a lot over the last few years. He's almost 4 now and I can get him to calm down enough to discuss things rationally. Sometimes not. Many times what triggers him is his litter sister doing the EXACT things he used to do to his older brother....like breaking apart train tracks, ripping a page out of a book, etc....but he LOSES it. Finally, I turn it into something funny and then he's ok with it, especially once I tell him that he used to do the exact same thing. Our 3 year old gets offended a lot and he often is not always things that someone did something intentionally. I have to show him the other side of the coin.
The other thing is, is that our 1st son is SUPER gregarious, very sports-oriented, will talk to anyone about anything. Our 3 year is not. Totally different personality and tempermant. He wanted to do soccer, so we signed him up. He was fine until more kids showed up, then he said he was done. He took him to taekwondo, where he's literally been, with our older son, since he was 4 days old - 3x/week. We got him there and he sat there looking at the ground. He finally came over and said he didn't want to do this. I was ok with that. I never push the kids to do anything, but rather, regarding chores give them impetus to do what I want on their own terms....especially with men, it had to be their idea! LOL
Maybe see what he wants to say, "I forgive you" is huge and if he really doesn't, then you are training his to say things he doesn't mean. I get why you are doing it, but maybe when that happens, remind him that we are not perfect - none of us - and have him come sit down with you until he can cool off. We use the words "mean" and "nice" a lot in our house. When he is ready, then he can go over and make nice. Our son needs to be alone to cool off and process it. When he is ready, then we quicly discuss what happened and i ask him , "Can you go over and kiss your sister and tell her that you love her?" His usual response is yes and then they kiss each other and ironically, BOTH of them say they are sorry and then both of them say it's ok. Our daughter is only 22 months.
Good luck....he'll be fine....and so will you. Parenting is certainly the hardest job I ever signed up for! The job description is constantly changing and once you get everything right, a new challenge arises. Keep breathing. Sounds like you are doing a great job.