A timeout can also a good way to allow emotions to cool (on both sides) and facilitate a better discussion. Each of my kids tends to "lose it" in a different direction (anxiety, sadness/crying, anger), and it's definitely important to support them as they work on getting their emotions back under control. This is going to look different for every kid though. My daughter needs space, one of my sons needs physical contact, the other needs a lot of verbal reassurance. It's important to help them find the ways they can work through their "big feelings" in a healthy way.
Your last point is critical. My father had a lot of strengths as a parent, but one thing I _never_ saw him do was apologize (not to me, not to my siblings, not to my mother). It's one thing I have really focused on "doing better" (something my father frequently encouraged me to do)